# Why Sleeping Beauty Is Lost in Time

The famous Sleeping Beauty problem has divided probability theorists, decision theorists and philosophers for over 15 years. Puzzle columnist Pradeep Mutalik claims to have discovered the source of confusion.

Mrzyk & Moriceau

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Editor’s note: In January, we ran an Insights column about the much-debated Sleeping Beauty problem. Now, our puzzle columnist Pradeep Mutalik claims to have discovered why this problem is so polarizing. In the spirit of experimentation, we will be inviting a panel of experts to weigh in on whether this insight adds any new clarity to the problem. Think of it as an informal Polymath-style “peer review.” Readers are also invited to share their opinions in the comments section below.

The famous Sleeping Beauty problem has polarized communities of mathematicians — probability theorists, decision theorists and philosophers — for over 15 years. In the puzzle, the fairy-tale princess participates in an experiment that starts on Sunday. She is told that she will be put to sleep, and while she is asleep a fair coin toss will determine how the experiment is to proceed. If the coin comes up heads, she will be awakened on Monday, interviewed, and put back to sleep, but she won’t remember this awakening because of an amnesia inducing drug she is given. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday, again without remembering either awakening. In either case, the experiment ends when she is awakened on Wednesday without being interviewed.

A monthly puzzle celebrating the sudden insights and unexpected twists of scientific problem solving. Your guide is Pradeep Mutalik, a medical research scientist at the Yale Center for Medical Informatics and a lifelong puzzle enthusiast.

Whenever Sleeping Beauty is awakened and interviewed, she won’t know which day it is or whether she has been awakened before. During each awakening, she is asked: “What is your degree of certainty that the coin landed heads?” (“Degree of certainty” is sometimes expressed as “belief,” “degree of belief,” “subjective certainty,” “subjective probability” or “credence.”) What should her answer be?

This simple mathematical problem has generated an unusually heated debate. The entrenched arguments between those who answer “one-half” (the camp called “halfers”) and those who say “one-third” (the “thirders”) put political debates to shame. In my column introducing the problem, I compared it to a Necker cube, the popular visual illusion that can be perceived in two completely different ways. But while most people can flip quite easily between the two views of the Necker cube, halfers and thirders tend to remain firmly rooted in their view of the Sleeping Beauty problem. Both camps can certainly do the math, so what makes them butt heads in vain? Is the problem underspecified? Is it ambiguous?

It’s hard to believe that this simply stated problem has remained open for over 15 years. It doesn’t seem to be underspecified — both camps feel confident that they have solved it. This suggests some deep ambiguity that causes smart people to disagree vehemently. In my solution column, I discussed some dichotomies in the two camps: Halfers count experiments, while thirders count awakenings; halfers calculate from the experimenter’s point of view, thirders from Sleeping Beauty’s. But these are mathematical techniques that both camps know how and when to use. When halfers and thirders reach different conclusions, it is clearly not a matter of mistakes in calculation: They must, in effect, be solving two entirely different problems.

This point — that the problem is ambiguous and both sides are correct — has been made by several people, including our own prizewinning reader Josh, who cited a paper by Berry Groisman. Groisman showed that there are two interpretations of the problem that are both consistent under standard probability theory. I agree, but this doesn’t explain why both halfers and thirders are so strongly convinced that only one side can be right. This phenomenon occurs, as Groisman suggests, because this quarrel is not about mathematics. Rather, it is a fight about two subconscious ways of understanding the problem’s statement, each linguistically valid and intuitively appealing. The fact that there are linguistic issues implying two different propositions here was pointed out by reader Nathan Gantz in his comment on our original Sleeping Beauty column. I show below that these two divergent approaches arise from different construals, or meanings, of the phrase “landed heads,” which refers to an event that happened in the past. These two meanings arise in human speech and writing whenever anything is expressed about a different time, and specifically whenever the past tense is used. In everyday communication we can figure out the correct meaning from context, but in the Sleeping Beauty problem both meanings are available for people to latch onto at an intuitive level.

To reveal these two meanings, let me add a small detail to the story:

Imagine that when the coin was tossed on Sunday, it was mounted on a brass plaque in the position it landed in, so that the result of the coin toss can be checked at any time. This plaque is kept in a locked safe in Sleeping Beauty’s room.

Certainly this act cannot make any difference to the logic of the actual problem. But it primes our intuitions, enabling us to see that the original question can be interpreted in two ways:

Meaning 1:  The Action Interpretation

“What is your degree of belief that the coin landed heads?” = What is your degree of belief that the coin landed heads in the act of tossing? (Imagine the coin being tossed.)

Note that the belief, though current, is about a previous event: The verb “landed” is in the past tense. Whenever a past event is evoked in speech or writing, the listener or reader has to decide how much of the event’s background is relevant. Sometimes, a phrase referring to the past requires the listener to “import” the event’s background without the speaker’s explicitly saying so — a “frozen past tense.”[1] It’s like a photo taken when you were 10 years old, showing your old house in the background, even though you’ve changed a lot since then and the house is gone. Here’s a question about the past that requires this kind of implicit background importing: What is your belief that my friend the rock star spent a full year’s pay on his first guitar? This question refers to your belief about the money my friend was making when he bought the guitar, not what he makes today. After all, he is a rock star now, 20 years later. In a similar way, the first meaning of the Sleeping Beauty proposition imports its background act, which can be intuitively accessed by invoking the image of the coin being tossed. It refers to the probability that the coin landed heads when it was tossed: obviously one-half.

Meaning 2:  The Property Interpretation

“What is your degree of belief that the coin landed heads?” = What is your degree of belief that the preserved coin in the safe is showing heads now? (Imagine which side is showing on the coin in the safe.)

Here the past-tense predicate “landed heads” is used as a way to describe a property previously gained by the coin you are referring to today. You are no longer concerned with how likely it was to have landed heads or tails when originally tossed, but only with the likelihood that the coin being referred to now, the preserved coin, shows heads and therefore “landed heads” at some time in the past (as opposed to landing tails). The same past-tense verb construction is used, but now it does not import the background action.

As an example, consider that until a few months ago,[2] six of the nine U.S. Supreme Court justices were graduates of Harvard Law School and the other three were graduates of Yale Law School. If I had asked you then, “What is your degree of belief that a random U.S. Supreme Court justice attended Yale Law School?” you would probably have answered one-third. Here, the question is obviously not intended to import the background of the event: I was not asking you about the odds that the justice chose Yale Law School out of all the law-school choices available at the beginning of his or her career, but rather, I was using “attended Yale” as a property of the person within the current group of justices, as opposed to the opposite property, “attended Harvard.” Similarly, the second meaning of the Sleeping Beauty proposition does not import the past action, but merely looks at heads as a present property, previously acquired. It can be intuitively accessed by invoking the image of the coin on the plaque in the safe, and refers to the probability that the coin is showing heads now, supposing you had to bet on it. This probability could be anywhere from 0 to 1. In this case, there are three possible cases where Sleeping Beauty could be examining the coin on the plaque: A Monday when the coin shows heads, a Monday when the coin shows tails, and a Tuesday when the coin shows tails. Therefore the probability of it showing heads is one-third.

For each real world question similar to the Sleeping Beauty question, context makes it clear what is expected: whether we should invoke the background of the event (the friend’s previous salary) or merely use the acquired property (the justice’s law school affiliation). However, the wording of the question in the original problem provides no context to force one interpretation or the other. Both are up for grabs — and boy are they grabbed tightly by the two different camps! To be fair, language understanding usually occurs automatically and subconsciously, so few of us question the ready-made interpretations that our minds settle on.

Halfers, I suggest, consciously or subconsciously find the first interpretation more salient, use it in their modeling, and come up with a value of one-half. Thirders subconsciously prefer the second interpretation, base their calculations upon that, and come up with a value of one-third.

How can this be? Isn’t the coin in the safe the same one that landed heads or tails on Sunday?

Let’s ask the sophisticated and intelligent princess Sleeping Beauty, who is well versed in the natural arts and sciences such as linguistics, math and science (not to mention fauna and flora). Let’s catch her at the time of her interview with the experimenter’s assistant at one of her awakenings.

Experimenter’s Assistant: What is your degree of belief for the proposition that the coin landed heads?

Sleeping Beauty: That question is ambiguous: It can be interpreted in two different ways. Do you mean my belief about the likelihood of heads in the act of tossing the coin on Sunday, or do you mean my belief about the likelihood of heads being shown on the preserved coin in the safe?

EA: But the coin in the safe is the same coin that was tossed on Sunday, and it shows the same result.

SB: Yes, but you can have a different belief about the probability of heads in the act of tossing a fair coin (which is always one-half at the time of tossing) and the probability of the same coin showing heads some time later.

Let’s suppose that while walking on the seashore, I see 15 coins, 10 of which show tails and five of which show heads. Perhaps a boy who was on the beach before me took away half of the coins that came up heads because he liked that side. No matter how many coins I gather, I always find two showing tails for every one showing heads. Half of the coins are, to me, lost in space. Now my level of belief that a new coin I encounter will show heads and therefore “landed heads” at some time in the past is only five out of 15, or one in three. I can only base my belief on the clear-cut and reliable statistics of the coins I encounter. Maybe one day, I’ll find the boy’s stash of coins that landed heads — and if I do, my expectation that there are equal numbers of coins that landed heads will change to one-half — or maybe I never will. Notice that we use the verbs “landed” or “came up” in two senses: for the act of tossing the coin, as in “the coin just landed heads,” and for the act of finding it later, as in “here’s a coin that shows heads and therefore landed heads sometime in the past.”

Here’s a different situation. Imagine I have a specific kind of double vision: The only objects it affects are coins showing tails. When a coin shows heads, I see it as one coin; when it shows tails, a strange optical effect makes me see double. I actually saw only 10 coins, five showing heads and five showing tails. But my strange affliction, which is unknown to me, causes me to see heads and tails in a ratio of 1-to-2. Hence I have an expectation of one-third that any new coin I find will show heads, and therefore landed heads. If I find out later that I have this condition, I can correct my erroneous, but previously valid, belief. Of course, knowing that these are all fair coins, I never waver in my belief that they originally landed heads one out of two times in the act of tossing.

Now imagine that I find the same 15 coins, but I’ve entered a time warp without knowing it, and five of the tails I saw were the same ones I had seen before. All the coins showing heads somehow escaped the time warp. Now half of the heads are lost in time, or you could say that the tails are doubled in time. Again, my belief that a new coin I encounter will be heads is, validly, one-third. If and when I come out of the time warp, and realize it, I will change my belief back to one-half.

I don’t want you to think that such distortions necessarily reflect some kind of error of judgment. It may be that half of all coins that landed heads self-destructed on landing so that the ratio of coins that I find reflects the existing ratio in the world. As long as I am unaware of any systematic errors, I therefore have to trust the ratio that I find as the basis for my belief in the probability of the coins I am likely to find.

Thus, distortions in time, space and perception that I am unaware of, or coins that exist for different lengths of time, or any systematic process that alters the frequencies of the two coin-toss results differentially, can alter the relative frequencies I reliably find. All these processes influence my valid belief regarding the proportion of the coins I am likely to encounter that landed heads.

Let’s return to your question, which is actually two separate questions. The first question is: What is my degree of belief that the coin landed heads in the act of being tossed? This value, of course, was one-half on Sunday, and will remain one-half until I actually find out what happened. I am a true halfer about this.

The second question — what is my degree of belief that the preserved coin is showing heads now — is the same as saying,What is my degree of belief that the coin in the safe landed heads sometime in the past?” On Monday and Tuesday, it is one-third, because I am in a time warp with half the heads lost in time. I am definitely a thirder in response to this question when I am awakened on these two days.

Olena Shmahalo/Quanta Magazine

When I emerge from the experiment’s time warp on Wednesday, the value of my belief in heads will return to one-half, because the two different interpretations will coincide. Then I expect that your boss, the professor, will tell me how the coin actually landed. At that time, my belief that the coin landed heads will settle on exactly 0 or exactly 1.

EA: Wow, all that sleeping must be good for the brain! Ouch, my head hurts. I think I’ll take some of the amnesia drug I’m about to give you…

That’s all there is to it, halfers and thirders. The rest, as they say, is just plain math.

After attaining the above perspective, I can honestly say that I can clearly see both views of the Sleeping Beauty Necker cube, and can readily flip from one view to the other. Now that you’ve heard the argument, here are three questions to answer:

1. Can you now disentangle the two different propositions implicit in the Sleeping Beauty problem?
2. If you were previously a committed halfer or thirder, can you now see the validity of the other point of view?
3. Do you think that this discussion throws any new light on the problem?

As usual, the best comment will earn its author a Quanta Magazine T-shirt. If a comment offers an idea that enhances this argument, it may earn the commenter an invitation to be a co-author if this argument is submitted to a journal for publication.

[1] Note that, as a couple of language experts including Steven Pinker of Harvard University have told me, this is not a phenomenon specific to a particular language. How much background we need to assume for a given statement is a universal problem and is usually resolved by context.

[2] The numbers in this example have been rendered obsolete by the death of Justice Scalia. I have kept the original numbers because of the relevant ratio they generate.

Note: This column was updated on April 8, 2016, to mention a reader comment that brought up linguistic issues. The columnist elaborates on the issue of assigning credit in the comments below.

• halfer all the way says:

I thought the root of the problem was thirders erroneuosly use the causality/dependency flow backwards.
Let's say she is a thirder, and the experiment lasts for the rest of her life. Now, because she is a thirder, she is forced to assess the degree of certainty of the coin landed heads as essentially zero. will that be rational?

• halfer all the way says:

I forgot to clarify my variation of the experiment: if the coin lands heads she will be awaked monday, if the coin lands tails she will be awaked every day for the rest of her life, not just monday and tuesday.

• Michael says:

I find the purported dichotomy confusing. SB is being asked what odds she will offer that the coin in the safe is 'heads', given her current state of knowledge (which includes knowing the rules, and that she is being interviewed). This appears to concord with 'property 2' defined above, i.e., the degree of belief 'now' (rather than 'then'). However, even so, she should not offer 1:2 odds, as is claimed, but 1:1 odds. 'Thirders' will lose money on average if they are willing to offer 1:2 odds.

There are indeed three possible scenarios that can actually occur under the stated conditions, which may be labelled as (H,Mon), (T,Mon), (T,Tue). The fourth possible scenario, (H,Tue) cannot occur under the stated conditions. If the experiment is repeated many times, then one must find
p(H) = p(H,Mon) + p(H,Tue) =1/2
p(T) = p(T,Mon) + p(T,Tue) = 1/2,
i.e, that the result is H on half the runs, and T on the other half. Otherwise the coin was not a fair coin. Note that p(H,Tue)=0 in the above equations, since the (H,Tue) scenario cannot occur.

To assign a reasonable degree of belief q to H, at the interview, means that Sleeping Beauty is willing to offer (or accept) betting odds of q:1-q on a heads result, confident that she will make a net zero payoff on average. This is the 'no Dutch book' definition of probability, and is useful in resolving the current problem. In particular, it means putting your money where your mouth is, rather than making various guesses as to 'ambiguities'.

Note that she is not offering odds on what day of the week it is, nor is she being asked for such odds (although she can take all aspects of her current knowledge into account when formulating her odds). The problem ask only what odds she should offer, when she is interviewed, for heads vs tails.

So, what value of q should she (or a bookie) assign? Well, she must be willing to offer $q to make a bet that pays her$1 if the coin in the safe is H, and to pay $1-q to make a bet that pays her$1 if the coin in the safe is T, such that she believes she will come out even. But the average payoff is then given by, adding up over the four scenarios,
P = q p(H,Mon) + q p(H,Tue) – (1-q) p(T,Mon) – (1-q) p(T,Tue)
= q p(H) – (1-q) p(T)
= (1/2) (2q – 1),
using the earlier calculations. This only vanishes for q=1/2. Hence, Sleeping Beauty will lose cold hard cash, on average, if she offers any other odds than 1:1.

Note in the above that it doesn't matter what probabilities are assigned to p(T,Mon) and p(T,Tue), or even that p(H,Tue)=0 in the statement of the problem. Under the given conditions, she simply has no knowledge about the particular day that is relevant to the odds of H vs T.

• Pace P. Nielsen says:

I think there is still a problem with your solution. Under the "property interpretation" of the puzzle, you seem to privileges all three states as equally likely for Sleeping Beauty to encounter. "But isn't that right?" you might ask. I would argue it depends on how you "measure" each state.

Suppose that the experimenters tell Sleeping Beauty that they will give her a dollar at the end of the experiment if she guesses the coin's face correctly. Since they are using the "property interpretation" they want to reward correct guesses of the *actual state* of the coin that's locked away. To punish wrong guesses, they tell her that they will remove any winnings for any wrong guess.

So they really do want to know about the true state of the locked-away coin. They want to test her belief. Of course, she wants to maximize her payout.

Is her best bet to guess 1/3-H and 2/3-T? Or is it better to guess 1/2-H and 1/2-T?

The first strategy gives an expected payout of 7/18. The second strategy has an expected payout of 3/8, which is less than 7/18.

So doesn't this tell us that the thirder's position really is the one to take here? Actually no, because both strategies are not optimal! In fact, given any 0 <=p <= 1 we find that a p-H and (1-p)-T strategy has an expected payout of 1/2 – p/2 + p^2/2. This is maximized when either p=0 or p=1. Thus, her best strategy is to decide (before the experiment) to pick one or the other–which seems to point towards a halver's position.

One possible objection to this set-up is that Sleeping Beauty is not revealing her belief about the actual coin, she is trying to maximize her payout.

I would actually agree with this objection. But here comes the punchline. I would argue, likewise, that the Sleeping Beauty in Pradeep's interview above is similarly maximizing a payout, completely unrelated to whether or not the locked away coin is heads or tails. She is "measuring" each of the three states as being equally likely to have happened, and giving her probabilities accordingly.

But Sleeping Beauty has another way of thinking about the universe. She could view the coin flip as determining two possible futures, each with equal likelihood. In one future, she will only be woken up on Monday. In the other future, she will only be woken up on Tuesday. When asked "What is the state of the locked away coin?" she doesn't care which actual day it is, and she doesn't care about getting the answer correct as many times as possible. She only cares about the coin's side. They are both equally likely.

Thus, it seems to me the solution to the Sleeping Beauty problem isn't about properties. Instead, it is one about "degrees" or "measurements". How are we measuring probabilities?

• Pace P. Nielsen says:

Of course when I wrote "only woken up on Tuesday", that should have been "woken up both Monday and Tuesday". Sorry!

• Luca says:

Thanks Pradeep for the stimulating discussion.
I find myself on the halfers side.
My main objection to thirders is why should SB believe the three awakenings (HM, HT, TT) are equally probable. The fact that she doesn't know which one of the three awakenings she is experiencing doesn't mean she cannot take into account their (different) probabilities. I think the situation is clearer if we consider a loaded coin with head probability 0.99. Do thirders still believe the probability is 1/3?

• Mayank Kejriwal says:

I enjoyed the article, and having had some experience with machine learning, I want to suggest a different way of putting forth the arguments. The difference between the thirders and the halfers is simply one of attributing the 'belief' of Sleeping Beauty to either the prior probability of the coin in the absence of any evidence (i.e. before waking up) for the latter, or the posterior probability in the presence of evidence (i.e. after waking up). In this framework, I believe that the 'rational' course of action is to take the new evidence into account (since we know Sleeping Beauty did wake up and was interviewed) and my bets lie whole and soul with the thirders. Probability works in strange ways sometimes. In the face of evidence, nature herself suggests a degree of belief must necessarily be updated; in nature, a mere act of observation is enough to lead to several intriguing paradoxes owing to quantum mechanics (EPR paradox). Are the halfers going to be so bold as to refute nature?

I do not mean to dispute what was written earlier in the article about there being two perfectly intuitive ways of looking at this issue, but intuitive is not the same thing as rational, and we have enough evidence from cognitive science to demonstrate the difference. Given to choices, then, are we not bound to favor the rational option? Let me frame the 'value' of a thirder solution in a way that a utilitarian would appreciate. Sleeping Beauty is told to place a monetary value on one solution over the other. If she is rational, she will take her evidence into account and choose 1/3 as the answer (and hence make an economic profit in the process).

So in conclusion, to answer your three questions, yes, I can disentangle the two propositions, I can sympathize with the halfers (but will not yield to them on any rational account) and while I believe the discussion was fascinating, the fact that there is evidence settles the question that the answer is 1/3. Let the chips fall where they may.

• A1philosopher says:

1. My belief, whether I am SB or not, that the coin landed heads is 0.5 since it was a fair coin toss with a random outcome.
2. As SB, am I in the state (H, M), (T, M) or (T, T)? If I didn't think about it I'd say that the probability of each was 0.333; but these are not equally likely distinct random outcomes. Since I have been anaesthetised I have no evidence as to which state obtains. Therefore I have to go back to the probability tree and find that p(H, M) = 0.5; p(T, M) = p(T, T) = 0.25 [since p(T, M or T) = 0.5].
Reasons for believing something need justification. Absent empirical evidence, probability provides a means of justification. But the fact that SB may be awake in three different scenarios does not mean that they are all equally probable so we need to examine the implications of each. If she is awake on Tuesday it is certain the coin landed tails. But it is impossible to determine the day of her awakening. However, if awake on Tuesday she must also have been awake on Monday (p = 1). And if she was awake on Monday, then the coin must have landed H or T – whichever, probability 0.5.

• Michael says:

It is interesting to see that several of us have argued that the question can be resolved by having Sleeping Beauty put her money where her mouth is, and confirm what her subjective probability is, by placing a corresponding bet at each interview (note that what someone conducting the experiment thinks is irrelevant to her subjective probability). Yet despite some agreement in this regard, not everyone agrees on what the average winnings/losses will be (and most simply assert a value).

Since one can easily simulate the experiment, and the bets, there is only one correct set of odds. I argued previously that they were 1:1. However, I now realise that I was incorrect – because SB is forced to bet twice when tails have been thrown.

In N runs, ~N/2 runs will be heads, and there are ~N/2 corresponding bets (on the Mon). The remaining ~N/2 runs will be tails, and she will have to bet ~N times (once on each Mon and Tue). So, in the N runs she has to make a total of about
N/2 + N = 3N/2
bets. If she bets heads each time, she will therefore only win ~N/2 bets out of a total of ~3N/2 bets (i.e, she will only win on the ~N/2 bets where heads was thrown). Hence, her subjective probability, corresponding to her odds of winning, is
p = (N/2) / (3N/2) = 1/3.
So, with this correction to my previous post, I am now a thirder 🙂

In regard to the questions
(1) The difference between halfers and thirders appears to lie in the number of bets one considers. If a person is simply betting on the outcome of each coin toss, then N bets are made on N runs, and the odds are 50:50. But SB is not: she is forced to bet only once if the outcome is heads, and twice if the outcome is tails. Hence she makes ~3N/2 bets on N runs, and it turns out her odds are then 1:2.
(2) I am a committed subjective probabilist. The question asks for SB's subjective probability, not anyone else's, and so (assuming she is rational), the answer is found via finding what her odds of winning are, not anyone else's. This leads to her being a thirder, as per above.
However, of course, an experimenter making just one bet per run will be a halfer.
(3) It does for me, since I incorrectly calculated the odds the first time 🙂

• Cathal says:

Suppose that if the coin comes up heads SB is woken on Monday as usual, but if the coin comes up tails, SB is woken instead on both Tuesday and Wednesday (sleeping through Monday). Each time she wakes up the experimenter asks "what day do you think this is?"
The experiment is repeated at the start of every week. What would her best answer be?

• eJ says:

Pradeep asks us to consider "my degree of belief" in the Boolean propositions:
* Proposition P1: the coin landed heads in the act of being tossed.
* Proposition P2: the coin in the safe landed heads sometime in the past.

P2 is too vaguely stated. Most coins have close to 100% chance of having landed heads /some/ time in the past, whether or not they're currently locked in a safe! You can't fix this: the dichotomy won't boil down to belief in two distinct propositions about the coin alone.

You would do better to ask: "given experimental set-up E, what is my belief that the coin is showing heads?", distinguishing two cases thus:
* E1: you wake up in one of two "worlds" (or timelines) determined by the coin flip
* E2: you wake up in the only "world" there ever was, with each interview equally likely

Underpinning those are "self-location" assumptions: the first is the self-sampling assumption (SSA, https://en.wikipedia.org/wiki/Self-sampling_assumption); the second is the self-indication assumption (SIA, https://en.wikipedia.org/wiki/Self-indication_assumption).

That's the crux of the dichotomy: the experimental set-up (E1 vs E2), not two ways of describing the same coin (P1 vs P2).

• Tom says:

SB determines the following before going to sleep for the first time:

1. 100% probability she will be awakened on Monday
2. 100% probability the Mon “awakening” will be erased from her memory
3. 50% probability she will be awakened on Tuesday
4. There will be a fair tossing of a coin on Sunday.
5. Immediately upon waking she will know it is either Mon, Tue or Wed
6. Sometime soon after each wakening, she will be given new information. Specifically, she we learn with 100% certainty whether or not today is Wed.
a) If she is asked “What is your degree of certainty that the coin landed heads?”, she knows with 100% certainty that today is either Mon or Tue
b) If she is told the experiment is complete, she knows with 100% certainty that today is Wed

Our initial focus will be on SB’s knowledge immediately following the question: “What is your degree of certainty that the coin landed heads?” – Here we go:

1. She can now locate herself in time within a 48 hour window – Somewhere between Mon morning and Tue night. Again, this is new information. The window narrowed from 72 to 48 hours.
2. Clearly (to me anyway), her “degree of certainty” needs to cover the entire 48 hour window.
3. In essence, from SB’s perspective, Mon & Tue are entangled and must be viewed as a single entity until she falls asleep again.
4. There is a 2/3 probability it is Mon. Also, if today is Monday, there is a 50% probability that the coin landed heads.
5. There is a 1/3 probability it is Tue. Also, If today is Tuesday, there is a 0% probability that the coin landed heads.
6. P(Mon, Heads) = 2/3 * 1/2 = 1/3
7. P(Tue, Heads) = 1/3 * 0 = 0
8. P(Mon, Heads) or P(Tue, Heads) = 1/3 + 0 = 1/3 “degree of certainty”. (SB’s Answer)

Random Thoughts

1. There is a 100% probability all Tuesday awakenings will be erased from SB’s memory. This does NOT have any bearing on the experimental outcome because all interviews take place before this particular memory is erased.

2. On Wed, AFTER she learns the experiment is a complete, if anyone asks her “degree of certainty”, it should now only include the Wed time window and the answer is 1/2.

3. After each “awakening” and before she learns whether today is Wed, she can locate herself in time in the 72 hour Mon, Tue, Wed window. I.e. Mon, Tue and Wed are now entangled. This leads to the following “degree of certainty” calculation:

a. P(Mon) = 2/5, P(Mon,Heads)= 2/5 * 1/2 = 1/5
b. P(Tue)= 1/5, P(Tue,Heads)= 1/5 * 0 = 0
c. P(Wed) = 2/5, P(Wed,Heads) = 2/5 * 1/2 = 1/5
d. P(Mon or Tue or Wed, Head) = 1/5 + 0 + 1/5 = 2/5 “degree of certainty” (Weird. I didn’t see that one coming!)

4. Consider an alternate experiment with the only one change; SB does NOT know her memory will be erased. In this scenario, at each “awakening”, she thinks its Monday and her “degree of certainty” is 1/2.

5. After working thru all this, I strongly believe the difficultly is NOT related so much to language but to the fact that SB’s memory will be erased and more importantly she knows her memory will be erased. This “forces” SB to entangle certain days during certain periods in the experiment.

• Hunter says:

I think Michael is right above.
Considering bets to be made by sleeping beauty about the coin, her "belief" depends on the rules governing when bets will be made.
If a bet is made at every interview: SB calculates the 1/3 answer.
If a single bet is made after the experiment: SB calculates the 1/2 answer.

It remains to be checked that talking about bets is an adequate description of what we mean by "belief."

• Michael says:

I thought I would have a go at answering some of the interesting variations posted by various commenters:

(A) Luca: "I think the situation is clearer if we consider a loaded coin with head probability 0.99. Do thirders still believe the probability is 1/3?"

No. In a total of N runs, SB is asked to bet once in ~0.99N of these (corresponding to heads), and to bet twice in ~0.01N runs. That's a total of ~1.01N bets. If she bets heads each time, her probability of being correct as N becomes arbitrarily large is therefore
p = 0.99N/1.01N ~ 0.98.
More generally, if the probability of throwing heads is w, then the probability of SB correctly guessing heads is
p = w/[w+2(1-w)] = w/(2-w).

(B) Cathal: "Suppose that if the coin comes up heads SB is woken on Monday as usual, but if the coin comes up tails, SB is woken instead on both Tuesday and Wednesday (sleeping through Monday). Each time she wakes up the experimenter asks "what day do you think this is?"
The experiment is repeated at the start of every week. What would her best answer be?"

Her best answer is "Whatever day you think it is" 🙂 However, if she is not allowed to get away with that, but has to bet on the choices of Mon, Tue or Wed, then there is no "best" answer: she will win 1/3 of the time no matter what her strategy is. In particular, if the coin is heads, which it will be in ~N/2 weeks of N runs in total, then she will bet once. If it is tails, which it will be in ~N/2 of N runs, she will bet twice in that week. So, during the N weeks she has to make a total of ~3N/2 bets. Further, ~N/2 bets will fall on each one of the three possible days. Hence, if she bets on, say, Tue each time, she will be correct with probability
p = (N/2) / (3N/2) = 1/3,
and similarly for the other days. It follows that even if she bets by randomly picking each of Mon, Tue and Wed with respectively probabilities x,y,z, then she will be correct with probability
(1/3) x + (1/3) y + (1/3) z = 1/3.

(C) halfer all the way: "Let's say she is a thirder, and … if the coin lands heads she will be awaked monday, if the coin lands tails she will be awaked every day for the rest of her life, not just monday and tuesday. Now, because she is a thirder, she is forced to assess the degree of certainty of the coin landed heads as essentially zero. will that be rational?"

I think she would notice she was getting old and decrepit after a few years, if it landed tails 🙂 So, I'll vary the variation a little: Suppose she is wakened on Mon if it lands heads, and that she is wakened on D consecutive days if it lands tails. Consider what happens in N runs of the experiments (it can be a different SB each time, if you like). In ~N/2 runs the coin is heads, and she is only asked to bet once. In ~N/2 runs the coin is tails, and she is asked to bet D times. So, the total number of bets she will have to make over the N runs is approximately
N/2 + D N/2 = (D+1)N/2.
If she bets heads each time, she will therefore be correct with probability
p = (N/2) / [(D+1)N/2] = 1/(D+1).
Note as D increases, she should therefore rationally assign a smaller degree of certainty to heads. Indeed, if she (or an observer) decided to be a 'halfer all the way', and therefore 'rationally' offered to receive $1 if it is heads in return for paying$1 if it is tails, each time she is interviewed (1:1 odds), then after N runs she can expect to make a total of
N/2 – (D+1)N/2 = – DN/2 dollars,
i.e., a large loss!

• Michael says:

Oops, in variation (C) in my previous comment, the 'total expected winnings' calculation, for a 'halfer' accepting 1:1 odds, should have been
N/2 – D N/2 = -(D-1)N/2,
i.e., an average loss of (D-1)/2 dollars per run rather than of D/2 dollars per run.

• Brian says:

It's just a matter of framing the question. The thirders and halfers are answering different questions (because the problem statement asks about "belief", which is ambiguous).

Here's a variant: I secretly flip a fair coin and you write down two guesses. If I see that the coin is heads, I disregard your second guess. If I see that it's tails, both of your guesses are valid. Of course, you're aware of all the rules before we start.

There's two questions we could answer here… 1) What's the likelihood that any individual guess is correct? 2) How can you get the most correct guesses?

The answer to the first question is 50%, as any single guess is independent of any other. The answer to the second question is more complicated because of the removal of one of your guesses after the event has taken place. That removal is a dependent event and affects the probability of the outcome, thereby resulting in an optimal strategy of guessing "tails" every time (the coin toss is truly 50/50, but if you always guess tails, you have a 2/3 chance of being correct overall).

The same is true of the SB problem – the two perspective examine two different things, where one is an independent event (the coin flip) and the other is a dependent event due to how the coin flip affects the rest of the experiment.

Both perspectives are valid because the problem is vague and ambiguous enough to support both scenarios.

• Davide says:

I totally agree with Michael's argument (april 2): if anybody had to bet at awakening, tails would be the most reasonable choice, and I bet also halfers would bet this way.

However, this is not completely fair: the game asks for 1 real numbers between 0 and 1, while this answer only provides an element out of a set of 2 elements, say {H,T}, which is a much poorer information.

The bet should then be placed by choosing an amount of money to put on H and another amount to put on T, and a payback should be fixed on sunday for the two options, in order to let SB choose the strategy that maximizes her income. Also, to be fair, SB should know whether who decided the paybacks is a thirder, a halfer, or an oracle.

• Sabeth Häublein says:

In the comments I read one objection to the thirder position I want to comment on. The objection went like this: if one would take an unfair coin with a probability of 99% that heads come up it would intuitively become apparent that the thirder position cannot be held. The objection concerns the thought, that there must be a higher probability of the event Monday/Head than Monday/Tale or Tuesday/Tale – respectively, p(M/T)+p(T/T) must be equal to p(M/H). I will show, however, that the objection is not valid.

Four the sake of my explanation I will use more accessible numbers.
Imagine the coin is unfair, so that p(Head) equals 3/4 and p(Tale) equals 1/4. If Sleeping Beauty wood for both cases only be awakened on Monday, the probabilities see evident (3/4 for Head and 1/4 for Tale). If, however, we add the awakening on Tuesday, we have to double the probability for Tale. It cannot just be divided in two equal parts, as we add one step to the experiment. Adding another step changes the relation within the system: We now have 3/4 +1/4 + 1/4, which is more than one. Having adjusted the calculation we end up with 3/5 for p(M/H), 1/5 for both p(M/T) and p(T/T).
(All of that can be done with the 99% example as well – though the simplicity of the calculation would be screwed: 99 parts and 1 part which has to be doubled. Now our new whole equals 101 parts which means that the new probability for p(M/H) would be 99/101, for p(M/T) 1/101 and the same for p(T/T). )

The outcome is that it definitely would be more probable that it was Heads and therefore Monday. Nevertheless, important for the discussion is the slight change in the relation.

Too now go back to the original idea with a fair coin: If Sleeping Beauty would be awakened only on Monday, the probability for Heads would be 1/2 as for Tales. If we as the additional awakening on Tuesday for Tales we double the probability for Tales and therefore have p(M/H) = 1/2, p(M/T)=1/2, p(T/T)=1/2. Adjusting the calculation we get p(M/H)=1/3 – as for the rest.

As a conclusion, we can say that the objection doesn't hold. What we intuitively consider as more probable changes it's probability during the process! It still is more probably for the princess to guess Monday if the coin was unfair but the clue about our problem is that the possible events remain equally probable.

• eJ says:

I see no evidence that SB philosophers find themselves tricked by linguistic ambiguity into Groisman's "setup of coin tossing" or "setup of wakening". Plenty of SB commentators recognise both "setups", explicitly aligning to one and rejecting the other. Whilst the path to either setup may muddied by language, the eventual choice of which to take is typically made with unswerving determination.

If we replaced SB's question with the version concerning a preserved coin in a safe, I am quite sure we'd see the same Halfer/Thirder tribalism. Nothing in the main article persuades me that the coin in the safe equates to Groisman's setup of wakening; in particular there is a hole in the argument where Pradeep asserts there are three possible cases (not two) and implies that those three have equal probability. No interpretation of "landed heads" does that for me.

I think what's really at stake is the proper mathematical treatment of Credence (not in quotes, as I don't wish to imply it's down to linguistics again). It's often treated as a subjective probability, amenable to Bayes Rule and a sound basis on which to place bets — this seems to align to the Thirder position. But need it be so?

Author’s Notes:

Formal Summary of the Argument

I have tried to elucidate the precise reason for the disagreement between halfers and thirders regarding the Sleeping Beauty problem and thus to uncover the reason why this seemingly simple problem is so polarizing. I now firmly believe that the disagreement is not about probability theory at all, but is traceable to a linguistic ambiguity in the problem statement. Here is a more formal summary of my argument:

1. Entangled propositions: The Sleeping Beauty (SB) problem statement contains not just one problem proposition, but two entangled ones. The two propositions concern SB’s degree of subjective certainty about: a) heads having landed in the act of tossing— this is the action interpretation proposition (AIP); and b) heads showing on the preserved state of the coin (hypothesized to be mounted in a plaque kept in the safe) on awakening—this is the property interpretation proposition (PIP). The correct answer to the AIP is 1/2, while that to the PIP is 1/3.

2. Ambiguity due to past tense construction: The AIP and PIP result naturally from two equally valid linguistic interpretations of the problem statement, which are based on the ambiguity of past tense constructions that are universal to natural language. Specifically, whenever anything is expressed in the past tense, the hearer (or reader) has to decide whether the background or original action of the event under consideration is being referenced (AI) or merely its last past result treated as a present property (PI). In normal communication this is easily decided by context. (It remains possible that there may be languages which demand that such ambiguities are always explicitly resolved: In such a language the entangled proposition structure of the Sleeping Beauty problem could not be expressed.)

3. Reason for polarization: As mentioned, normal real-world communication contains enough contextual information to allow us to resolve these ambiguities and pick one interpretation, whereas the context-free nature of a coin toss in a mathematical problem like the SB problem allows both interpretations to be available. The reason that this problem is so polarizing is that most solvers intuitively latch on to one or the other alternative propositions subconsciously and use it to solve the problem. Once they do this, they are too far along to notice the other interpretation. When halfers and thirders talk to each other, they are usually talking past each other. What the adherents of each camp have concluded about various aspects of their favored proposition is irrelevant to the other equally valid proposition. It’s no wonder that the other sides’s argument seem absurd — they are about different propositions.

4. Circumstances in which the AIP and PIP answers diverge: The degrees of certainty about the AIP and the PIP in case of any particular coin toss can vary from 0 to 1 depending on the subject’s state of knowledge and circumstances of the experiment. In most normal circumstances the answers coincide, but they are based on different kinds of information. The AIP requires only knowledge of how the actual coin toss turned out; the PIP, on the other hand, can yield different answers based on the experimental setup. Under unusual circumstances such as the amnesia of SB, processes unknown to the subject that affect heads and tails differently such as losses, duplications, distortions or differential survival of heads and tails can alter a subject’s reliably experienced or calculated statistics and therefore give her different, valid belief values about AIP and the PIP. Such a distortion occurs in the SB experiment: it can be compared to a time warp on Tuesday which includes tails but excludes heads.

5. Equivalence of Bayesian and frequentist reasoning: By treating the AI and PI propositions as different, the normal equivalence between probabilistic reasoning and frequentist reasoning is preserved. There is nothing in this problem to cause this equivalence to break, despite some speculations to the contrary: both techniques give exactly the same answers, as they are expected to do in simple calculations like this one. Any conclusion that can be reached by Bayesian reasoning can also be reached by counting frequencies in the entire sample space, or by collecting statistics. This foundational pillar of reasoning about probabilities remains intact, as it obviously should. Specifically, we can apply any of these equivalent techniques to the AIP and the PIP in the Sleeping Beauty problem, and they will all give the same answers: 1/2 for the AI proposition and 1/3 for the PI.

6. The solution of the AIP (Answer =1/2): The correct answer of the AIP—the probability of heads in the act of tossing the coin, given SB’s knowledge—is one-half. The argument for this was first given by Lewis (2001). It hinges on the fact that Sleeping Beauty gets no new information about the actual result of the coin toss (the AIP) in the experiment, so her original degree of certainty of heads (1/2) should be retained. She cannot do Bayesian updating for the AIP, because she simply has no information to calculate posterior probabilities. We could imagine slight variations of the experiment in which she does get such information: for example, if there were a 50% chance of her being woken up on both Monday and Tuesday on heads, and a 100% chance of being woken up both days on tails. This constitutes probabilistic information about the actual result of the coin toss, which is relevant to the AI proposition. This kind of information can indeed change SB’s degree of certainty for the AI proposition to a value anywhere between 0 and 1 depending on the actual numbers involved.

On the other hand, the PI proposition is affected by other experimental details besides the result of the coin toss: what matters in PI is the information about the likelihood of encountering heads on any given awakening. This is dependent not just on information about the result of the coin toss, whether or not that is available, but also the details of the experiment such as differential encounters with heads or tails, or differential longevity, or the other kinds of circumstances mentioned in Sleeping Beauty’s monologue above. On account of this, Lewis’s argument does not affect the result of the PI proposition. Note that the argument attributed to Nick Bostrom—that Sleeping Beauty does have new evidence about her future from Sunday: "that she is now in it," and that she gains the information that it is not both Tuesday and Heads was flipped— does not constitute information about the AIP, but only about the PIP.

7. Solution of the PIP (Answer =1/3): The correct answer to the PIP—the probability of the finding that the preserved coin shows heads—is one-third, the argument for which was first given by Elga (2000). The probability of Sleeping Beauty encountering heads on Monday – P(H/M) – is the same as her probability of encountering tails on Monday P(T/M). Also her probability of encountering tails of Tuesday P(T/T) is the same as P(T/M), because the former occurs whenever the latter occurs. This gives us P(H/M) = P(T/M) = P(T/T), which when normalized, gives a probability of heads = 1/3. Note that for PI, the assignment of probabilities 0.5, 0.25 and 0.25 to P(H/M), P(T/M) and P(T/T) respectively ¬– as done by A1philosopher here in the comments – is plain wrong for the PIP. Specifically it fails to take into account the fact that heads and tails are equally likely on Monday, so their probabilities have to be equal. These probabilities are caused by the circumstances of the experiment. The famous “Extreme Sleeping Beauty variant” in which SB is awakened every day for a million times on tails but only on Monday for heads, does indeed reduce p(Heads) for the PIP to 1 in 1,000,001. However it has no effect whatsoever on the AIP which remains 1/2. If the two propositions are not clearly separated in one’s mind, these large numbers could have an effect of shaking and misleading one’s intuition. Note that such extreme numbers should not be unsettling for subjective probabilities: after all, when you actually know the truth, subjective probability finally does settle on exactly 0 or exactly 1 — as far away from 1/2 as can possibly be imagined.

8. Is the ambiguity caused by something else? Many people (for instance, Brian and eJ here) have expressed the opinion that the reason for the ambiguity is that the problem is about “subjective certainty” or “belief” or “credence” which supposedly is inherently ambiguous. I submit that this view this mistaken. There exist very well-defined, completely unambiguous procedures that can be used to operationalize the degree of subjective certainty a rational agent may have in a given proposition. These are well described in the Wikipedia page on the SB problem: By offering a bet with well-defined odds; more elaborately by setting up a Dutch book; or by repeating the experiment many times and collecting statistics. Thus for example, if your degree of certainty about the probability of rain tomorrow is 1/3 then you will be willing to accept odds of more than $2 on a$1 bet, on the proposition that it will rain tomorrow. If you were offered lower odds than 2:1, you would bet on the opposite proposition (in such cases, a bet with well-defined odds is offered to the individual whose degree of certainty you want to test, and he or she is free to accept either side of the bet). If you want to remove the subjectivity of belief completely from the problem, you can just substitute this operationalization of certainty instead. Even when you do that, though, the ambiguity still remains. The results are different if SB is thinking about the AIP (which gives an answer of 1/2: see my figure below) or if she is thinking about the PIP (which gives an answer of 1/3). This proves that the ambiguity is about the proposition, not about what the definition of subjective certainty is.

To those who are still unconvinced about this point, I would encourage you to imagine any situation whatsoever in which you would invoke subjective probabilities, and see that the above operationalization works: It is completely objective and well-defined, and not just an arbitrary bet that can be changed at will. There is absolutely no basis to reject this standard procedure (De Finetti’s operational subjective view of probability) for the SB problem.

While a lot of commenters here have given correct views from the standard halfer and thirder perspectives, no one (apart from eJ) has even acknowledged the two different propositions I have described. Hopefully, the above explanation will provide some clarity. I will try and reply to individual comments later. For now I want to put up two illustrations that hopefully may help you visualize the two different propositions, and get a clearer picture of what I’m trying to say.

• eJ says:

Pradeep, your point 8 does not prove that the definition of credence is free of ambiguity. With its starting assertion of "very well-defined, completely unambiguous" credence-testing procedures, it's almost a circular argument to conclude that credence itself is well-defined.

Consider betting. We cannot offer SB "a" bet, in isolation, on Tails without bringing-in some other random factor to choose the betting day on the Tails time-line; and in doing so, we change the set-up. In offering potentially multiple linked bets, it is not at all clear that the odds/payoff should be related to "credence". We've argued about this before; you know there's ambiguity here.

– – –

But let's return to your coin-in-safe phrasing of the problem. I am intrigued to know if it persuades Halfers to reach a Thirder conclusion. Perhaps it does, and you're on to something (without having disproved credence ambiguity!).

To all Halfers — what should Beauty's credence be that the coin in the safe is showing heads now?

• rjt says:

Even after thinking about this post over several days, I don't understand how Pradeep's solution is different from Groisman's. The principle behind both of these solutions is that the original problem is ill-posed and that there are multiple possible interpretations. Pradeep expands on Groisman by claiming that the ambiguity is due to linguistic considerations. I don't think that this is true, however.

The opposite extreme of Pradeep's "coin on a plaque in a safe" reformulation would be to suppose that, immediately following the coin toss, the coin is destroyed. Disintegrated. Blown to smithereens. Now there is no plausible way for SB to interpret the question as referring to the status of the coin when she awakens. Yet, the thirder position is still plausible. This illustration suggests to me that the problem has nothing to do with "ambiguity of past tense constructions that are universal to natural language", as Pradeep suggested.

We can take this reasoning one step further by inverting the temporal order, and thus the causality, of the events. Suppose that SB is guaranteed to sponteneously awaken either once or twice. As usual, she cannot remember any previous awakenings, and does not know what day it is. If she awakens only once, then on Wednesday, the experimenter will place the coin with heads up. If she awakens twice, then on Wednesday the experimenter will place the coin with tails up. When SB awakens, she is interviewed. "What is your degree of certainty that the coin will be placed with heads up?"

This version of the puzzle uses no past tense or references to events in the past. But it is still subject to the diverging thirder/halfer interpretations. Let me be clear: I think the problem is due to ambiguity in how the question is interpreted, but this ambiguity is not due to linguistic considerations of past events.

• Devin says:

Thanks for this post. It makes clear that no matter how straightforward the solution is, humans will spend the rest of their lives passingly defending the beliefs they have identified with. It will be a great day when educated humans can agree on the analysis of the sleeping beauty story.

Credit assignment

The assignment of credit regarding who came up with a specific idea first, is always a very vexed one in a problem that has as voluminous a literature as this (as it is in science in general).

It has been pointed out to me that reader Nathan Gantz had invoked the idea of two distinct propositions inherent in the Sleeping Beauty problem in his comment in the original Insights column on this problem. Specifically Nathan, mentioned that

“Instead of asking Sleeping Beauty what her certainty is of heads or tails, we can ask her to guess which one occurred… Linguistically, it removes the distraction of a temporal shift in the halfer position…" in a comment on January 18 at 12:56 am.

This does implicitly get at the idea of linguistic ambiguity as being the source of confusion in the Sleeping Beauty problem.

I have revised the text of the column to give due credit to Nathan.

One of the things I hope to sharply delineate in this column is the question, “To what extent is my treatment of the problem original?” When I developed my argument, I was surprised to see that the past tense linguistic double interpretation was not mentioned or given the prominence it deserves in mainstream treatments of the problem. I believe that my clear-cut analysis of the two past tense construals, and my identification and characterizations of the circumstances in which the two propositions yield different answers as elaborated in the various scenarios sketched by Sleeping Beauty in her monologue above, are original. On the other hand, the two propositions have been clearly identified from the perspective of probability theory by Groisman, and my contributions are in the nature of support to this already well sketched-out position. Even this much is not certain: There is a voluminous scientific literature on this problem over 15-16 years that I have not yet completely examined, so it is quite possible that all I have said has already been discussed many years earlier by someone. It is with this in mind that I invite experts more familiar with this literature to comment on it.

Was I influenced by Nathan’s comment back in January? It’s possible, though I was a bit confused about his “being right on paper” part, and I did not comment on it as I did on several other reader comments. As I remember it, the idea of linguistic ambiguity was driven home to me forcefully when I was looking at some other comments at other blog sites written many years ago that also did not explicitly state it. All I can say for certain is that this aspect of linguistic ambiguity deserves greater prominence in treatments of this problem, and if it has any element of originality, I would like to submit a paper on it. I think that a wider and deeper understanding of this issue should put an end to the polarization that this problem continues to engender.

• Nathan Gantz says:

I hope assignment of credit never becomes a huge priority in STEM. But since we are on the topic, I just want to clarify two things. One, I can claim no credit for anything Pradeep has done beyond (at most) "identification" of the problem. Two, when I identified the linguistic problem, I specifically referred to it as "temporal." I think that a statement can be made in passing yet still be a direct statement. Sometimes the lemma is more valuable than the proof.

As Brian earlier states the halfers and thirders do seem to be answering different questions, or more to the point, that Snow White is answering the question based on two different motivations.

Outside of this experiment 50/50

Inside the experiment people seem to be debating these two variations
Halfer version:
Snow White is told she will get a $1 reward on Wednesday if at any time she answers correctly, in which case her choice is 50/50 as the payout is the same. Thirder version: Snow White is told she will get a reward on Wednesday of 1$ for every time she answers correctly. In which case if she wishes to maximise her potential payout she should say tails irrespective of her actual degree of belief that the coin landed heads.

Which is fine but the Snow White isn't asked "Did it land heads yes/no?".
If there is neither reward nor punishment nor accrual or accounting of correct answers then what should Snow White say?
I'm a halfer.. 50/50

• Lucas says:

The people who are introduced to this sleeping beauty problem, take the time to "process" it in the "mind," with the use of this "biological brain" thing we have are not "looking at two different problems." In fact all problems are "not different" and therefore we can only deal with this thy called "perspective." Depending on where the "observer" or "problem solver" "is standing" will determine what is "perceived." Yes, everyone knows this, though not everyone sees the significance of this. The reality is that both the "halfers" and "thirders" are both correct. It's interesting how we know of crazy s**t in this "existence" thing and yet understanding/accepting the concept that two answers are correct for one problem is acceptable.

With this, I point out the fact that if we try and remove perspective, then we try and remove the way we "know existence" this task is done in vain. So, the question comes down to when a coin is flipped, what are the chances it is heads. This portion is all answered within the boundaries of human perspective

• Laurence Reeves says:

I'm not a believer. I do think that most coins seem to be close to fair. As stated, the coin is definitely fair. It matters not when or how many times you wake me, if you ask me what credence I give to the coin, in the problem as posed, having been being heads, I will say 50%. But that's what you have told me the probability is. I'm a halfer.

However, no one seems to have modified the problem to be "Monday only" versus "Every day for seven weeks". I'm not a fiftiether – still a halfer.

If you attach any merit (or better still, cash) to whenever my "belief" is borne out by the actuality, then my largest payout comes with me saying that I "believe" the coin was tails. In my above variant, I'd even go so far as to say "I fervently believe the coin was tails". I would, of course, be lying.

• Hiter Peter says:

The linguistic solution is most probably wrong. While past tense is indeed ambiguous and has the described two interpretations (known fact in linguistics), there is no evidence that this particular distinction is relevant here. You need to show that the halfer/thirder ratio actually does change with a change of wording; it is not enough to ask if people feel the explanation plausible. Ask 100 people (who hear the problem for the first time) with wording #1, and another 100 people (who also hear it for the first time) with wording #2. My guess is that you would not see any meaningful difference in the halfer/thirder ratios. Note that many of those 200 might still feel your explanation plausible after you tell it to them. People are easily influenced by eloquent speakers on philosophical issues. You'd better be skeptical until you see actual data.

Why I think your explanation is wrong? Because I have my own resolution (with zero evidence to back it, of course)! Here it is:

Thirder perspective: Each wakening is a new bet. Betting tails every time will win her more money than betting heads every time. Can't possibly be 50-50.

Halfer perspective: On Sunday, she assigns a subjective probability for the coin, namely 50-50. Each time awake, she is allowed to change that belief, given the "new information" that she's been awaken (though amnesia prevents her from remembering any change). Should she make the change? Interview happens with 100% probability (namely, 50% heads/Monday + 25% tails/Monday + 25% tails/Tuesday), so being awaken is definitely something to be expected. Nothing unusual, no hints. No new information, therefore, no change in belief warranted.

So, as I see it, halfers model "belief", as it should rationally change when confronted with acquiring new information (no new information = no change in belief). Thirders model betting strategy, for a particular set of betting rules (more correct bets = hopefully more money). Of course, belief is used for optimizing behavior ("betting"), and while in a game different betting rules are possible, in real life nobody tells you the rules, and you should assume by default that two chances at shooting give you better odds than just one chance. In that case, betting tails 2/3 of the time is warranted.

In summary: my analyzing self is definitely a halfer, and my "belief estimate"/"subjective probability" will always be 50%. No linguistic argument can convert me. But for my get-the-job-done self that doesn't matter, I would just always bet on tails (like a thirder) if asked only a few times, and collect actual data if playing repeatedly.

Thus far, I seem to have failed miserably at even conveying the idea that there are two different propositions here, let alone my hypothesized linguistic reason for them ☹. A lot of the commenters continue to identify in the traditional way as halfers or thirders. My hope was to get many more of you to say “I am inclined to be a halfer\thirder, but I can see that there could be another possible way to interpret the problem statement.” Oh, well! I’ll chalk it down to the psychology of polarization, or as Devin nicely put it, “humans will spend the rest of their lives…defending the beliefs they have identified with.” If this can happen for a mathematical problem, what hope do we have to reduce polarization in politics?

So let me try one-on-one comments, starting with those who have given some indication that they do see two sides.

@Abaddon, Laurence Reeves, Brian, eJ and Hiter Peter

What I hear you saying is that, as far as belief is concerned, you are halfers. You recognize that in practice being a thirder will win you a bet offered every time you are awakened, but you are loath to call that a belief about the state of the coin. Fair enough. What this tells me is that you prefer the action interpretation of the problem statement, namely: “What is the probability of heads having landed in the act of tossing? The answer is obviously 50%.

So for a moment, answer eJ’s call and forget the original problem statement. Imagine that you never heard the problem this way at all. Imagine that right from the beginning, the only way the problem was posed to you was as follows: You are Sleeping Beauty and you have been awakened on Monday or Tuesday (you don’t know which) in a state of amnesia about possible past awakenings, and you are asked: What is your degree of certainty that the coin in the safe is showing heads now?

What would your answer be? You are not being asked about how the coin landed. You are asked about your degree of certainty or belief in a simple proposition — the probability of the coin in the safe being heads —just as you might be asked what is your degree of belief in the probability of rain tomorrow.

You cannot just duck and say that I have no belief about this particular proposition. If you can say about any proposition whatsoever, “I think there is a 1/3rd or one-half chance of rain tomorrow, or a 1/10th chance of this horse winning a race, or a 1/13 th chance of drawing an ace from a full deck of cards” then you can certainly pick a number to indicate your degree of certainty regarding this particular proposition.

I think you will have to say 1/3.

What this means is that you find the property interpretation unpalatable when the action interpretation of the original SB problem exists as an alternative. But in its absence, you are forced to take the same position.

For thirders, this interpretation of the problem question is perfectly natural, even when the action interpretation is offered as an alternative. It makes sense linguistically, and therefore, to them, it is an equally valid answer to the Sleeping Beauty problem.

Take a minute and see how you would answer the question about the Supreme Court justices in the main article, and see if you can see the linguistic parallel.

You do have a right to use the action or halfer interpretation of the SB problem. But can you now see that the thirders have the same right (to select the property or thirder interpretation)?

In fact, I find much to agree with what Hiter Peter wrote, even though he said I was wrong. He agrees that there are two interpretations as I described. He recommends that we should actually conduct the experiment and ask a couple of hundred people the SB problem with two different wordings. I agree, though there are methodological problems with this in people who have heard the SB problem before: We’ll have to choose people who have never been exposed to it. Finally, I also agree with Hiter that halfers probably have a strong model of belief, whereas thirders model betting strategy. I mentioned a couple of months ago in my previous column on this topic that I think thirders are probably thinking more pragmatically. I am not saying that it is the linguistics that makes you pick a side in the SB problem—that happens because of your modeling predilections, as we just discussed. What the two alternative linguistic interpretations do is that, no matter what choice you make, for whatever reason, you can find a readymade interpretation that fits your choice and convinces you that you have solved the problem as it was posed.

Cheers!

@eJ

Thank you for being open minded and floating your question to all halfers. I hope your call succeeds in getting responses though it will have to overcome the deep entrenchment of habit to make them change their answers.

No, I don’t think there is any ambiguity here. We did argue about it, but never resolved it. I think that, at bottom, degree of certainty or credence is a very simple concept that is completely captured by De Finetti’s operational subjective view—the existential bet as I once described it. There is nothing more to it. What’s much more important, in this problem and elsewhere, is the actual proposition whose degree of certainty or credence you are trying to determine.

The De Finetti procedure is certainly not circular. After all, a degree of certainty about a proposition is something mental and therefore private. The willingness to accept the specified bet is an operationalization that provides a behavioral test that lays bare for all the world to see what your degree of certainty is: If you do not do this then you can change your mind or wiggle out of it and there is no way to nail you down. It’s what people mean when they say “Put your money where your mouth is.” You can operationalize degree of certainty in some other way, but you will be hard pressed to come up with something else because this is the obvious bet for capturing it. And it is universal. You can apply it to any probabilistic proposition whatsoever.

Here’s how it applies to a halfer’s credence. On Wednesday, after SB has exited her unusual experimental conditions, she will be perfectly willing to accept a bet with odds greater than $1 for a$1 bet on heads. She will do this because she is a halfer on Wednesday. Conversely, the fact that she will do this indicates that she is a halfer on Wednesday. The bet reflects the degree of certainty; the degree of certainty reflects the bet she is willing to accept.

It’s that simple.

So here’s a challenge. If you don’t like this operationalization, come up with some other one that can be applied universally for any proposition such as the probability of rain tomorrow, and all the other examples I gave in my last post. Then apply it to the SB problem. Or conversely, use the definition of credence that you want to use in the SB problem for all these simple propositions that I described.

Regarding the SIA and the SSA, yes, they correspond to my two interpretations, the PIP and the AIP. However they are assumptions required in anthropic reasoning: we don’t need assumptions for the SB problem. We just need interpretations of the problem statement.

Consider this for a moment: both halfers and thirders are convinced that they have the right answer to the SB problem, which has a single statement. Doesn’t that show that they are interpreting that same statement differently? My analysis shows how that comes about: linguistically, both interpretations are valid. How else do you explain the conviction that both camps feel that they have solved the problem as posed?

@eJ (continued)

You mentioned that "We cannot offer SB "a" bet, in isolation, on Tails without bringing-in some other random factor to choose the betting day on the Tails time-line; and in doing so, we change the set-up."

We can indeed offer the existential bet at any time whatsoever. It reflects the credence at that particular time and no other. So SB's credence (under the PIP) is 1/2 on Sunday, 1/3 on Monday and Tuesday, and 1/2 again on Wednesday, when she escapes the experimental time warp. The bets she will accept at these times reflect these credences. Of course, credences can change in time. When the result of the coin toss is finally revealed, her credence goes to 0 or 1: she knows for certain what the result was, so she will accept a bet at any odds whatsoever on the correct choice and no bet at all on the wrong one.

@Michael

I think everything you have written after your retraction of the first comment is correct. I appreciate your refutation of the objections raised by Luca and halfer all the way.

However you have only considered one interpretation of the problem statement: the PIP, or what the coin in the safe is showing. To address the other one, consider what you, as a subjective probabilist would bet on Wednesday after SB is awakened but before she is told the result of the coin toss. It would have to reflect a probability of heads of ½. This shows that even as late as Wednesday, SB has no additional information about the AIP — the actual result of the coin toss — after Sunday. She is aware of this on Monday and Tuesday. So if you consider the actual result of the coin toss (the AIP), you have to be a halfer.

@Sabeth
I commend you too for your answer to Luca’s objection. But please refer to my comment to Michael above (paragraph 2) for the halfer perspective.

@Mayank
See my comment to Michael above (paragraph 2).

@rjt
My solution is not different from Groisman’s. It supports Groisman’s solution, provides the linguistic justification for it, traces the ambiguity back to the original problem statement, and offers an explanation for why this problem is so polarizing.

If the coin is destroyed, the “ghost” of the coin still exists — the experiment technician who interviews SB will still know how the coin landed and what face would show if it were still in the safe. That’s why the thirder position is still possible.

I do not understand your inversion of temporal order example. What is the probability of SB to awake spontaneously? If she has a 50% chance of awakening on Monday only, and a 50% chance of awakening on both Monday and Tuesday, then only the halfer position is still possible: The thirder position is impossible. So the ambiguity vanishes. Doesn’t that show that the past tense is necessary?

@Pace P. Nielsen
Your “other way of thinking about the universe” is precisely my AIP and corresponds to the halfer position — the probability of heads in the actual toss of the coin.

However, in your analysis of the PIP and the payout, you use a bet of “1/3H and 2/3T” etc. Bets are never made probabilistically: you always bet on one outcome or the other against definite odds. SB’s payout is indeed maximized when she bets on tails and not on heads — something you fail to mention. So your analysis does reflect that the thirder position is correct under the PIP.

@Pace P. Nielsen and Hunter
Re. measurement vs probability:
Specific kinds of bets (which I call “existential bets”) do indeed reflect probabilities — see De Finetti’s operationalization of subjective probability. Check out my point number 8, and my discussions with eJ above.

PS
Here’s a detailed description of how a bet can reflect a probability.

Simply, if your subjective probability of a proposition is 1/p, then rationally you would accept odds of (p-1):1 or greater in favor of it.

@eJ,

One last point.

Continuing my last post to you:
We can offer SB an existential bet on the AIP on Sunday and Wednesday and ascertain that she is indeed a halfer about the AIP. We cannot physically do it on Monday and Tuesday, because she is imprisoned in the experimental time warp. But she still has the knowledge about the coin toss (or lack of it) at the back of her mind, and can still answer a question when asked about it, as she does in her interview in the main article. As that interview illustrates, SB is simultaneously a halfer about the AIP and a thirder about the PIP.

"You are asked about your degree of certainty or belief in a simple proposition — the probability of the coin in the safe being heads." as stated, if this is a bet, and I win everytime I give the right answer, then yeah tails. For a tails bet 50% – 0$, 50% 2$, Heads 50% 0$50% 1$ assuming no change in bet on the Tuesday, and why would I when I have apriori decided I will always bet tails.

However I will concede, that if the question is
"What is your degree of certainty that this is the second time I have asked you about the coin in the safe" 1/3rd every time.

I said in my last post that we cannot physically offer SB an existential bet on the AIP: My reasoning was that it would contravene the stated experimental protocol to offer an extra bet. However, I just thought of a way in which it could be done.

The experimenter employs two assistants: one carries out the standard interviews of the SB experiment. The other assistant—let’s call her the special assistant—randomly chooses a time on Monday or Tuesday to wake up SB and carry out a single special interview outside the standard protocol and in addition to it. As before, the amnesia agent is given after this interview, so she forgets about it and it does not in any way interfere with the actual experiment. SB is told about this beforehand, and can readily recognize the special assistant.

At this special interview, SB is asked what odds she would accept that the coin landed heads. Now we have successfully transported SB outside the experimental time warp, and have made the AIP and the PIP coincide. SB’s answer is of course, odds greater than $1 for a$1 bet—reflecting that she is, in this situation, a halfer about both propositions.

• Wiley says:

"Credence" is another word for Bayesian probability, but mathematics of probability has been developed on an entire axiomatic basis that is independent of any interpretation. Once someone writes down the probability space, you may ask whether it is consistent, which is purely a mathematical question, i.e., whether it satisfies the Kolmogorov axioms. On the other hand, you may ask whether it is justified. To decide e.g., between two sets of consistent probabilities, one inevitably has to appeal to other fields outside of mathematics. This is the main reason that the sleeping beauty problem remains controversial among philosophers rather than mathematicians, evidently given the extensive literature in philosophy:
http://philpapers.org/browse/sleeping-beauty/

Some formality. Denote the background proposition (experiment setup):
B = "the fairy-tale princess participates in an experiment that starts on Sunday, …, the experiment ends when she is awakened on Wednesday without being interviewed";
and the events:
H = coin toss results head, T = coin toss results tail; Mo = today is Monday, Tu = today is Tuesday.

We can all agree that on Sunday, beauty has a prior credence of 1/2 for H. The question is her posterior credence during the experiment.

The paradoxical nature of the problem arises from the inconsistency of the following "intuitive" assignment of probabilities from beauty's perspective:
(proof is straightforward by applying Bayes' theorem)

(1) P(H and Tu) = 0; (2) P(Mo or Tu) = 1; (3) P(H | Mo or Tu) = 1/2; (4) P(H | Mo) = 1/2; (5) P(Mo | T) = 1/2.

Since the beauty knows B, the setup of the experiment, we cannot remove (1) and (2).

Thirders would like to replace (3) with 1/3. This is counter-intuitive because it is difficult to see how the awakening provides new information for the amnesiac beauty to update her posterior probability of H;

Halfers would like to replace (4) with 2/3. This is counter-intuitive as well due to the lack of new information: the beauty knew she would be awakened on Monday no matter what the coin toss outcome is.
(I have not seen this line of rebuttal from thirders here, except it was briefly mentioned when eJ states his double-halfer position. Halfers could defend by using an argument that is similar to Monte Hall problem, which depends on how exactly beauty gets the information of Mo)

Let me emphasize that being "counter-intuitive" is NOT necessarily wrong. One can check that the probability space resulting from either replacement is mathematically consistent thus valid. Is halfers' model "boring" and less useful in terms of betting? I would leave this to philosophers, but there is no "Dutch book" (known in financial mathematics as "arbitrage") against the halfers. The major issue of this type of betting argument is that the existence of the bets depends on the outcome of the bets (Bradley and Leitgeb 2006).

Personally I find giving up either (3) or (4) is equally unpleasant, so I tend to agree with double-halfers to give up (5). The price to pay is to give up Mo and Tu as valid events, because the propositions are so-called indexical due to the usage of "today", but this goes beyond the scope of the discussion here.

@Wiley,

Have you read Groisman's paper and considered my linguistic argument in detail? By clearly delineating two different propositions, you do not have to give up any of your 5 assignments: they simply get organized in two different groups corresponding to the two different propositions. I believe this resolves the paradox.

• Ian says:

"In this case, there are three possible cases where Sleeping Beauty could be examining the coin on the plaque: A Monday when the coin shows heads, a Monday when the coin shows tails, and a Tuesday when the coin shows tails. Therefore the probability of it showing heads is one-third."

SB (or we ourselves) could reason as above, but she would be wrong if she did. She, like us, knows about the fair coin toss, and that gives her extra, relevant information that she would be ignoring. The above "logic" hides the implicit (and invalid) assumption that, when multiple cases exist, they must all be of equal probability. That certainly can be the case, but most definitely need not be. And, here, it isn't.

SB cannot treat the 3 cases as equally probable, because she knows that the decision as to whether to awake her either once or twice will be based on a fair coin toss, and must take that into account when evaluating possibilities.

The coin toss was fair; the probability that she will be awakened once is therefore 1/2, and similarly the probability that she will be awakened twice is 1/2.

If she is being awakened only once, the DEPENDENT probability that she has just been awakened on a Monday is 1. The overall probability that she has been awakened on a Monday when the coin shows heads is therefore the product of the two: 1/2 x 1, or 1/2.

If she is at some point in the course of being awakened twice, it is equally likely that she has just been awakened on a Monday or a Tuesday (both will happen or neither). The DEPENDENT probability of each is therefore simply 1 divided by the number of awakenings – 1/2. The overall probability of EACH INDIVIDUAL case in which she has been awakened on a day when the coin shows tails is therefore the probability that the coin IS showing tails, times the dependent probability of the individual case: 1/2 x 1/2, or 1/4.

So, to correct the original, quoted logic:
"In this case, there are three possible cases where Sleeping Beauty could be examining the coin on the plaque: A Monday when the coin shows heads (probability one-half), a Monday when the coin shows tails (probability one-quarter), and a Tuesday when the coin shows tails (probability one-quarter). Therefore the probability of it showing heads is one-half."

(If anyone needs a Reductio ad absurdum, here's an attempt:

"Dissolute son-of-a-country-gentleman John Golightly has been found guilty of theft, and is waiting in the cells for the morrow, when the judge will hand down his sentence. Judge Swingumhigh is famous in such cases for dispensing death or transportation upon the simple toss of a coin.

"Reasoning as does SB, John's father has engaged the services of that renowned German doctor, Franz Mesmer. Herr Doktor Mesmer is to visit John in his cell, and place him into a deep mesmeric condition. The doctor has assured John's father that John will be able to attend court and receive sentence, but will remain in his state.

Once sentence has been past, it has been arranged that the doctor will visit John in his cell once more and awaken him from his trance. On awakening, he promises, John will remember nothing of the sentence handed down. Seeing an opportunity to aid his son, John's father has paid Herr Doktor Mesmer a substantial sum of money to agree that, if the sentence is the more minor one of transportation, the doctor will then place John back into a trance no less than eight times, each time awakening him without any memory of either the sentence or of any previous awakenings.

John's father is confident that, as in only one of one ten will John be awakened to be told of his impending death, he as a father has done all that anyone could wish of a father, by reduced the probability of his son being roused to be told of a death sentence from one chance in two to the much more promising one chance in ten – and that the rather generous sum he has promised Herr Doktor Mesmer is therefore money well spent.

John – knowing that his life still hangs upon the mere toss of a coin – is not entirely convinced that justice can so easily be cheated.")

• Ian says:

Ugh. There's nothing like posting a long comment (with glaring spelling and grammar errors to boot) and then realising it's wrong. (A.k. "How I flipped from being a 1/2-er to 1/3".)

This is about the difference between the base probability that the coin is heads, and the probability from SB's perspective that an individual experience will be heads. If the coin comes up tails, SB will always experience 2 tails events; if heads, only 1 heads event. And if the experiment were repeated many times, she would still, on average, experience roughly twice as many tails events as heads events (all, from her viewpoint, equivalent). And all events are, from her perspective, equal (even though she knows they're not). So – and even though she can calculate that she will actually experience a heads event with a probability of 1/2 each time the experiment is repeated – the probability that an individual experience is of a heads event is, from her perspective, still only 1/3.

• eJ says:

Just briefly, counter to Pradeep's "credence is a very simple concept that is completely captured by De Finetti’s operational subjective view", consider the Analysis journal article "When Betting Odds and Credences Come Apart: More Worries for Dutch Book Arguments" (http://joelvelasco.net/teaching/3865/leitgeb%20and%20bradley%20-%20FairBets.pdf). Unless you wish to dismiss that and its bibliography out of hand, you have to accept that credence can be contentious.

• rjt says:

"I do not understand your inversion of temporal order example. What is the probability of SB to awake spontaneously? If she has a 50% chance of awakening on Monday only, and a 50% chance of awakening on both Monday and Tuesday, then only the halfer position is still possible: The thirder position is impossible. So the ambiguity vanishes. Doesn’t that show that the past tense is necessary?"

In the example of the inverted temporal order, I was intending, as you guessed, for there to be a 50% chance for SB to awaken on Monday, and a 50% chance for her to awaken on both Monday and Tuesday. However, the thirder position is still viable.

As I see it, the same reasoning as in the standard SB puzzle can apply. From SB's perspective, each of the three possible awakening scenarios are equally likely, and in only one of those situations will the coin be placed heads-up. Therefore, her subjective probability that the coin will be placed heads-up is 1/3.

The halfer reasoning is also equally applicable. (I won't re-tread that ground.)

So, there is definitely something linguistic/interpretational going on, but it is not related to a reference to past events.

(The other reason that I am skeptical of the past events explanation is that the alleged "ambiguity" is not one that is recognized by formal logicians or semanticists, as far as I know. The majority of ambiguities in natural languages are easily resolved by context, and the ones that aren't are usually genuinely puzzling – people don't tend to have strong, immediate opinions that cannot be swayed. Instead, they're less confident about the interpretation and their understanding can easily be modified by additional information. The idea that thirders are spontaneously understanding a past-tense verb to refer to a present state is not consistent with the semantics of English. I think that Pradeep is poking in the right direction, but the past tense stuff is a red herring.)

• Larry Franz says:

I wasn't a halfer or a thirder before reading the article. Now I'm a committed thirder.

If you were to simply ask Sleeping Beauty whether a fair coin toss came up heads, she should say the odds were 1/2. Without any other information, that's the rational answer. But you're asking what odds Sleeping Beauty should assign, given the additional information she's been given about the experiment. Since today could be either Monday or Tuesday (as far as she knows), it's more likely that the coin came up tails. The fully-informed, rational answer she should give is 1/3.

So I think Sleeping Beauty should say this each morning: "Given how coin flips work, the odds are 1/2 that heads came up. But given how coin flips work and given what you've told me about this peculiar experiment, the odds are only 1/3 that heads came up. I can easily flip back and forth between the halfer and thirder positions, but why should I ignore the additional information you've given me? Taking into account what I know about the experiment, I must conclude that the odds are 1/3 that heads came up. If that makes me a thirder, so be it. Now where's my prince?"

• Rich says:

Luca said earlier on:

'My main objection to thirders is why should SB believe the three awakenings (HM, HT, TT) are equally probable. The fact that she doesn't know which one of the three awakenings she is experiencing doesn't mean she cannot take into account their (different) probabilities'.

This is a very interesting point, and I've seen a number of comments where halfers do not think the probabilities for the three awakenings are the same, while I do think they are the same. This seems to be a concrete difference between some thirders and halfers.

From my perspective, the situation is

– the probability of the awakening (HM) occurring = 0.5
– the probability of (TM) = 0.5
– the probability of (TT) = 0.5

These are the probabilities SB would estimate with knowledge that the coin is fair, before the experiment is started. There is no new information available to her later to change this assessment, so this is her best estimate at any point in the experiment.

These probabilities do not sum to 1, which is ok because they do not refer to a single event, but the three events seem to me to be equally probable. Can someone who disagrees explain their position?

From my perspective, when SB is awakened, she knows with probability 1 that she is in one of the three awakenings above, and that they are each equally probable, and therefore the probability that she is in any particular one of them is 1/3. As heads occurs in only one of them, and tails in two, this makes me a thirder!

• Alex says:

So is there a fifty percent chance that I am a thirder, it a one third chance that I am a halfer?

Really simple statement of the Groisman/Mutalik argument

Imagine that 100 fair coins are tossed, and as expected, about 50 come up heads and 50 tails, but subsequently some asymmetric process destroys or hides all the heads and only heads. Now you are certain that the coins were fair when tossed (probability of heads = 1/2), but you can also be certain that any coin you encounter in the future will be one showing tails (probability of heads = 0).

It is clear that the probabilities of these two events vary independently; one is dependent on the fairness of the coin, the other on both the initial ratio and the degree of asymmetry in what happens after tossing. Satisfy yourself that whatever the probability of heads in tossing, you can always imagine subsequent processes that can result in any probability whatsoever of heads being encountered later.

This is the crux of the idea: that there are two propositions in the Sleeping Beauty problem:
1) the probability of heads when the fair coin is tossed, which is obviously 1/2. This is the proposition modeled by halfers.
2) the probability of heads encountered later, which thanks to the experimental conditions that are asymmetric between heads and tails, is 1/3. This is the proposition modeled by thirders.

The tossed vs encountered distinction is exactly the same as the action interpretation vs property interpretation distinction I discussed earlier. Refer to the main article above to see how the same problem statement can mean tossed if understood one way and encountered if interpreted another way.

Once you accept that there are two valid interpretations, the solution of the Sleeping Beauty problem can be expressed in one sentence:

A fair coin is tossed: The probability of it landing heads when tossed is ½; the probability of Sleeping Beauty encountering heads later, thanks to the asymmetric experimental conditions, is 1/3.

And that, in a nutshell, is all there is to it.

More Replies

@rjt

In your description, there are only two possibilities, both of which are equally likely: either SB wakes once (heads) or she wakes twice (tails). So the probability of heads is clearly half. There is no other possibility.

When I first encountered the SB problem I was a thirder, so I understand the thirder mindset. There is absolutely no way I can conceive why anyone would be a thirder for your version.

As far as your point about ambiguities is concerned, you are correct about the strong immediate opinions that people have. Here I agree with Hiter Peter: the strong opinions are formed by other things, such as strong intuitions about beliefs perhaps, and/or a procedural vs pragmatic mathematical style. The two alternative past tense interpretations are just available to support whatever conclusion a person has already reached based on personality and preference, and help support the feeling that the question has been addressed.

By the way, the past tense verb refers to a property acquired in the past and it is fully compatible with English (and other language) semantics. See my example of the Supreme Court Justices. What is the chance that any one of them went to Yale? It is simultaneously a property acquired in the past, as well as a label used to distinguish it from its opposite label —which was also acquired in the past.

@Ian
At least you reached the right conclusion. Well done!
But realize that you have calculated the probability of “encountering” heads. There is still the other proposition — what did the coin actually do when tossed? The probability of this is ½ because SB does not know what actually happened on the coin toss. (See the tossed/encountered distinction in my “really simple statement.”)

@Rich
You have definitely identified a key sticking point for many halfers. Your analysis is correct except for a minor detail: HM and TM are two mutually exclusive parts of one event space (the Monday event space), and as you rightly state, both have probability = 0.5. The TT event is separate and always follows the TM event (which we know has p=0.5) with probability 1. So all the three are indeed equiprobable as you state, and normalize to 1/3 when we treat them as one event space which SB’s amnesia forces us to do.

Halfers who do not accept this are either:
a) Making an elementary error in the application of probability theory; or
b) Are intuitively convinced that this is impossible because there is no new information that warrants the change from ½ to 1/3.

As my tossed/encountered distinction shows, they do not have to be worried about the second point. Information is only required to change the coin’s tossed probability, not the encountered one, which can change quite independently.

@Larry
As in my reply to Rich, your answer is correct for the heads “encountered” probability. The heads “tossed” probability is, and remains ½ until SB learns the actual result of the coin toss.

The point of all this is to show that there is no incompatibility in the thirder and halfer arguments. The same person can simultaneously be both a halfer and a thirder.

@Alex
Both are correct. There is a fifty percent chance that you are a thirder, and a one third chance that you are a halfer. Plus, as this column shows, there is as an additional 2/3 chance that you are both a halfer and a thirder. There—you are now complete 🙂

Cheers!

• Michael says:

I appreciate the "really simple statement" post, and the one-sentence conclusion that
"A fair coin is tossed: The probability of it landing heads when tossed is ½; the probability of Sleeping Beauty encountering heads later, thanks to the asymmetric experimental conditions, is 1/3".

However, I do not agree this justifies concluding that there is any ambiguity, about the probability Sleeping Beauty should assign, in the problem as it is stated in the original post above.

In particular, the problem as given at the top of this page states:
"During each awakening, she is asked: “What is your degree of certainty that the coin landed heads?” …. What should her answer be?".

First, for there to be any meaningful answer, she must be prepared to use it to inform any decisions she might make. Otherwise, the answer "-8" or "the square root of minus 1" are equally good answers.

Second, since she cannot distinguish between any of the three possible awakenings, she must be prepared to give the same answer at any of the awakenings.

Now, lets suppose that p=1/2 is a rational answer for her to give at each awakening This implies that it is rational for her to accept the bet "I will pay you $15 if it is heads, but you will pay me$10 if it is tails", at each awakening. Why? Because she will reason that her average winnings per awakening will be
1/2 $15 – 1/2$10 = $2.50, and hence she will be ahead on average by at least$2.50 per experiment, no matter whether there are one or two awakenings.

However, p=1/2 is not in fact a rational answer, because if she accepts the bet at each awakening she will actually lose on average. Why? In N runs of the experiment the coin will be heads in N/2 of them, with just one awakening, and so she will win a total of $15N/2 in those N/2 runs. The coin will be tails in the remaining N/2 runs, with two awakenings each time, and so she will lose a total of$10N in those remaining runs. Hence, her average winnings per experiment are
$15/2 –$10 = – $2.50, i.e, she will in fact on average lose money if she gives the answer p=1/2. Thus, given that the answer she gives at each awakening is to be operationally meaningful for her, for that awakening, the answer p=1/2 is an irrational one. In fact, the only rational answer, satisfying the first and second criteria above, is p=1/3. There is no ambiguity in the above example: if she is a halfer, then she will lose if she acts on her belief. I therefore think the notion that there is some obvious ambiguity in the problem is not correct. There may still be quite subtle ambiguities (which I won't muddy the waters on the main point by discussing here), but certainly not a "time" ambiguity as suggested in the post. Sleeping Beauty knows the conditions of the expt before it is ever given, and we know them too. Hence the rational answer for her to give at each awakening can be evaluated at any time, before, after or during the experiment. • rjt says: @Pradeep Thanks for responding. Although this is ostensibly a problem in probability theory, since the paradox appears to be due to people's diverging interpretations, it's really a psychology / behavioral science issue. So I'm afraid that you saying that you cannot conceive "why anyone would be a thirder" for my version of the problem doesn't quite cut it. I think the thirder position is equally as valid for my version as for the original version. This could be an error in my understanding, but I don't really see what makes the two versions so different. (I'm curious if other commenters can get the thirder position for this scenario too.) To be clear what I was meaning in my earlier comment (although I'm pretty sure you understood): the three possibilities are: SB wakes up for the first and only time (heads), wakes up for the first time of two awakenings (tails), wakes up for the second of two awakenings (tails). From the perspective of SB, each of these situations are equally likely and she cannot distinguish them. As far as I can tell (and I'm very happy to be corrected), this reasoning is equivalent to the thirder position for the standard SB problem. With regards the past tense interpretation: yes, the past tense is used to refer to events of the past, which may or may not cause properties that persist into the future. But the ambiguity that you are claiming requires a past tense construction to refer to a present property while ignoring the past, which is not a feature of English. Again, to be clear: I think that you have correctly identified that the problem with the SB problem is a linguistic or interpretational ambiguity, presumably due to a mismatch of referents between problem-solvers, and that there is no real incompatability between the positions and one can consistently hold both positions. However, I don't think that it is due to an "ambiguity of past tense constructions that are universal to natural language". • Pradeep Mutalik says: @eJ, I went through the Bradley and Leitgeb (2006) paper that you linked to. I find no evidence to suggest that they define credence as anything other than simply another word for Bayesian or subjective probability, as Wiley suggested. Yes, they do consider complex propositions in certain situations where they claim that credence does not follow Dutch book arguments, but this once again shows that it is the propositions that are important, not the definition of credence, for which you can merely substitute the words “subjective probability.” Thus, Sleeping Beauty’s question, wherever it appears, can be phrased as “What is your subjective estimate of probability that the coin landed heads.” This way, we banish the words “belief “ and “credence” and keep the discussion in the realm of probabilities. This is important because these words are distracting—anything that has even a whiff of “belief” has all kinds of other, sometimes sacrosanct, connotations in people’s minds. It’s no wonder that the motivation for the Bradley-Leitgeb paper was the Sleeping Beauty problem. Why do you think the definition of subjective probability, which can be operationalized using de Finetti bets for a million other problems, suddenly fails for the Sleeping Beauty problem? I’ll tell you what I think. This paper was written before the Groisman paper, so the authors had no idea about what I’m now calling the tossed/encountered distinction. Obviously, if you try to analyze what you think is a single proposition but actually is two different independent ones, you’re bound to find a whole lot of contradictions and non-conformities. I have started reviewing the classic papers on the Sleeping Beauty problem and the whole field is a mess. People were going through all kinds of contortions, questioning textbook staples such as the axioms of probability, the equivalence of Bayesian probability and frequentist statistics, and so on. Even Bostrom makes ridiculous appeals to intuition (as in the extreme Sleeping Beauty variation) to make points that in retrospect are completely unjustified because the premise is about the “encountered” proposition and the conclusion is about the “tossed” proposition, which can clearly be shown to vary quite independently of each other. Have you ever had the experience of holding a hopelessly knotted ball of string, and by chance tugging at the right places to find that it suddenly fell apart and separated into two clean strands as if by magic? That’s exactly what I felt when I really “got” Groisman’s idea. I think that it is absolutely revolutionary – it can completely dispel the fog of confusion and polarization that surrounds this problem, and explain why it existed in the first place! All the rules of probability work just as they do elsewhere without these convoluted exceptions specially invented for the Sleeping Beauty problem. I really want to make this crystal clear to you, so I am constantly trying to find better ways of expressing it. But it does require entrenched thirders and halfers to forget, for a few minutes, everything they have read or concluded about the problem and start afresh. So I’d like to plead with you to really make an effort to really understand this new paradigm – I’d be happy to provide any clarifications. You won’t be disappointed 🙂 Look, eJ: You know that I was as passionate a thirder as any. But now, as you can see in the comments, I feel compelled to defend the halfer point of view just as passionately. Moreover, I can now clearly see that both Elga’s thirder argument and Lewis’s information argument are both absolutely correct and do not conflict with each other at all. Can you? If not, wouldn’t you like to? Cheers! • Pradeep Mutalik says: @Michael If Sleeping Beauty interprets the question to be "what is the degree of certainty that the coin landed heads, in the act of tossing: i.e. how did the coin actually fall?" rather than a betting question, then the answer has to be one-half. After all, she has no knowledge of how the coin actually fell, because the asymmetric experimental conditions were not based on anything that the coin did. As someone who thinks in betting terms, you can see this clearly by considering how she would bet on Wednesday (when the asymmetric experimental conditions are removed and both questions coincide). On Wednesday she would have to bet as if p(Heads)=1/2, and nothing about this probability has changed from Sunday to Wednesday. Now this question is completely different from the betting question "What is the coin in the safe right now?" You are absolutely right that the answer to that question is 1/3 on Monday and Tuesday. The two answers are answers to the same question understood differently. I tried to make this clear in the Sleeping Beauty dialogue in the article. You can also look at the figure in my summary post: if SB is imagining how the coin fell in the coin toss, the answer is 1/2. If she is imagining what the coin is in the safe (the "encountered" coin) then the answer is 1/3. As you can see from my 100 coin example, the tossed/encountered answers can be completely different. When she bets, she has to use the "encountered" version of the question which is 1/3. • Michael says: @rjt You have asked for opinions/support re thirder position for a "temporal inversion" variant of the problem: "Suppose that SB is guaranteed to sponteneously awaken either once or twice. As usual, she cannot remember any previous awakenings, and does not know what day it is. If she awakens only once, then on Wednesday, the experimenter will place the coin with heads up. If she awakens twice, then on Wednesday the experimenter will place the coin with tails up. When SB awakens, she is interviewed. "What is your degree of certainty that the coin will be placed with heads up?" You clarify in a later post that she is woken once during the expt, with probability 1/2, and twice in the expt, with probability 1/2. In particular, you and @pradeep disagree over whether p=1/3 is a rational response, which is interesting, as it illuminates the issue. I agree with you that p=1/3 is indeed rational for this version, just as it was for the original version, for essentially the same reasons. Further, I think that p=1/2 is not rational. In particular, consider what happens if she bets on heads on each awakening, in N runs of the experiment. In N/2 runs, she is only woken once, and the coin will be heads, so she wins N/2 times. In the other N/2 runs, she is woken twice, and the coin will be tails, so she loses 2 N/2 times. Note there are 3N/2 awakenings in total. Hence, the probability of winning is p = (N/2)/(3N/2) = 1/3. One can also show, as per the explicit example I gave in my previous post, that if she is prepared to bet based on the answer p=1/2, then she is going to lose money. Hence p=1/2 is not a rational answer. So, I agree with you that temporal ordering is not relevant to the original problem. What is relevant is that number of awakenings is correlated with the coin toss outcome, and, since she has to assign the same probability on each awakening (since she cannot distinguish them), the weight or probability that she assigns to the coin outcome depends on this correlation. There are some interesting subtleties re the connection between probability and betting that can be raised, but Pradeep seems quite happy to make the usual connections in various responses, at least up until now (thus, one would typically assume, for example, that there is a record of the bet, and any payoffs are made after each run: variants of this can muddy the waters, but it is the standard scenario that payoffs for games are made this way). • Michael says: @Pradeep "As someone who thinks in betting terms, you can see this clearly by considering how she would bet on Wednesday (when the asymmetric experimental conditions are removed and both questions coincide). On Wednesday she would have to bet as if p(Heads)=1/2, and nothing about this probability has changed from Sunday to Wednesday." SB is not asked to bet on Wednesday, nor to give her probability on Wednesday, in the problem as originally stated. She is asked to give her probability at each awakening on Mon and/or Tue. Hence stating that on Wed she should say p=1/2 is simply not part of the problem. Indeed, now suppose the experiment is modified, and she is also asked on awakening on Wednesday what her probability of heads is. She doesn't know it is Wednesday, as she has amnesia. So now there are 2 awakenings if the coin is heads, and 3 awakenings if the coin is tails, and she cannot distinguish any of them. The rational probability for heads that she should assign on each awakening, is then p = 2/5, by the same arguments as before. Again, if the experiment is modified so that she is not awakened at all until Wednesday, then there is only one awakening if the coin is heads, and one awakening if the coin is tails. The rational probability for heads she should assign on each awakening is then p = 1/2. However, in either modification, we are not discussing the original problem, so it is not surprising that the answer is different. • eJ says: Pradeep, I'm happy that you've reached the "bargaining" stage of the Kübler-Ross model for the death of Thirdism 🙂 Perhaps I'm still stuck at "denial". The core new idea of your approach has become tricky to discern amongst the many posts here. If it's the thing about frozen past tense, in particular that the coin-in-a-safe rephrasing forces a 1/3 answer, then the scientific thing to do is to test it (I guess with a poll) … how about with a captive audience of college students? I've not seen such a thing in the SB literature — it would be an interesting contribution. In reference to your Property Interpretation, you say it's "[the probability that] the coin in the safe landed heads sometime in the past" or, more recently, it's "the probability of heads encountered later". Those natural-language statements are imprecise, as they don't specify the sampling mechanism. There will be people out there that interpret both in the Halfer sense. Something for the survey to tease out (let's not argue it further here)? Maybe you'll find Halfers split into militant and malleable sub-species. Finally, Groisman appears to be a Thirder: "I agree with thirders that her credence on awakening should be 1/3". His argument appears to be that the poor darling Halfers haven't quite grasped that the puzzle is about credence at wake-up time, not credence a priori. He doesn't recognise that Halfers can understand the puzzle perfectly well and yet still have a non-Thirder model of SB's thinking at those crucial interviews. Your contribution will be much more interesting than that if you are making a claim not about *our* interpretation of the *puzzle*, but instead about *SB's* interpretation of the *interview* question. • Jacob says: I think the article gets the problem essentially right. One way to think about it in my opinion is to change the problem slightly to one I feel is easier to interpret. Consider that me and a friend are sitting in two separate rooms. I have a coin, a die and a piece of paper with a heads and a tails column. I flip the coin and if it lands heads I chalk up an "H" in the heads column walk over to the other room with the coin in my hand. If the flip comes up tails I chalk up a "T" in the tails column. I then roll the die, and if the die shows six I walk over to the other room. If the die shows anything but a six, I flip the coin again, and repeat the process. Now the probability that I hold a coin that came up tails when I walk into the other room is just 1/2*1/6=1/12. But obviously the probability of me flipping tails is 1/2. So when I ask "What is the probability that this coin came up tails?", the question (as the author stated) can be interpreted in two ways. Either my friend can answer the question as a statement about the probability that the coin I'm walking in with came up tails, in which case the answer is 1/12. Or he can interpret is a statement about the long run ratio of "H"'s and "T"'s on my paper. If we were to repeat this experiment indefinitely I really would be walking in with a coin that came up tails 1/12 of the time but the "H"/"T" ratio really would be 1/2. It's just that he doesn't get to see most of the "T"-tosses. So it all depends on what probability you are referring to. • tab says: Based on the information Sleeping Beauty is given on Sunday, it is trivial for SB to determine the day of week during each awakening. SB surreptitiously mark herself in some distinct way during at each awakening. Examples: pinch herself hard enough to create bruise, bite the inside of her left/right cheek, bite a particular nail, scratch her skin, etc. Therefore, her answers are: – Monday (no mark): 1/2 – Tuesday (one mark): 0 – Wed (based on one or two marks respectively): 1 or 0 Question for “halfers”; Do you agree with this analysis or do you still maintain the question boils down to “What is the definition of a fair coin toss?” • George says: I haven't read all the comments, so I apologize if this thought has already been given. I find myself in the halfers camp because the set of coin tosses is a singleton set. Even if we interpret the question as about her experience, that is the number of times she wakes, then since she is the one stating belief and she can't access this information she must go with the probability associated with the toss, because it is a single toss, not the proportion of an imagined set of experiences. The problem of the thirders, maybe, is that they too quickly project the experiment into the future, imagining many Sunday's passing. The proportion of tosses that will be heads on the days sleeping beauty is woken approaches 1/3 as the number of tosses increases. This cannot happen with a singleton set of outcomes. The probability remains 1/2 for each toss, but the proportions change in a predictable way for the set of her experiences once the number of tosses being considered is greater than one. She will be right to say heads has a one third chance, because we are double counting tails, in effect. This proportion cannot emerge without multiple tosses. The probability for her woken days emerges out of an extended experiment, but the probability for each toss remains 1/2, which is in fact what SB is guessing at. • Pace P. Nielsen says: @Dradeep, Thank you for responding. First, it seems that you are incorrect that "bets are never made probabilistically". For instance, if you flip and coin and ask me to bet on the outcome I am completely free to flip another coin myself and make my bet the outcome of my own coin-flip. Moreover, assuming your coin is fair, my probabilistic bet will have the same chance of rewarding me as any other method. So, indeed, some logical bets *are* made probabilistically. Moreover, there are situations where this type of probabilistic betting results in better payoffs than choosing one result or another. This may be important in a moment. Second, you assert that "SB's payout is indeed maximized when she bets on tails and not on heads — something you fail to mention." I failed to mention it because in the set-up I gave, which I thought conformed to your PIP model, this is not true. It appears that my set-up fails your PIP model for some reason. So please help me to figure out what that reason is, as I explore some options below. I think you see the scenario as follows. You might say that if we run the experiment over and over, about 1/3 of the time the correct answer to the question "what is the actual state of the coin locked in the safe" is heads, and the other 2/3 of the time the correct answer is tails. I agree with this, but only if we weigh each of the three states with the same weight. So, my question to you, is: What is it about the PIP interpretation (i.e. what is it about the actual state of the locked away coin) that determines SB wants to weigh each of the three states the same? Note that the answer cannot be "she wants to maximize the correctness of her answers" because that has nothing to do with the actual state of the coin. • A. McLeod says: First of all I’ll admit that I’d never heard of the Sleeping Beauty problem before today, and that I haven’t read any of the literature; I’ve read just this article and its comments, along with Pradeep’s above-linked article from January. But I have to say I find myself a committed “thirder” at this point, and I don’t see how one could take the “halfer” position without ignoring part of the available information in the problem. Here’s how I’ve come to think about it: Suppose the experiment is run as described above, except that the experiment may start on any day of the week, so that instead of Monday and Tuesday we have “day 1” and “day 2” (you’ll see why in a moment). Now let us suppose that during the experiment, the researchers run a call centre with two operators, A and B. At any point on day 1 or day 2, operator A may call a random mathematician, explain the SB experiment, and then say “I am giving you the following information: it is either day 1 or day 2, and the coin has been flipped. What is your degree of certainty that the coin landed heads?” At any point on a day when Sleeping Beauty wakes up, operator B may call a random mathematician, explain the SB experiment, and then say “I am giving you the following information: it is either day 1 or day 2, the coin has been flipped, and Sleeping Beauty woke up today. What is your degree of certainty that the coin landed heads?” (Note that the call centre operators do not tell the mathematicians anything about the operations of the call centre itself, so the mathematicians have no information about the probability that will they be called by operator A versus the probability that they will be called by operator B.) A mathematician called by operator A should answer “1/2” since the question she was asked is equivalent to “A fair coin has been flipped; what is your degree of certainty that the coin landed heads?” A mathematician called by operator B should answer “1/3” because she is stating the conditional probability that the coin landed heads, given that it’s a day that Sleeping Beauty wakes up. As has been extensively discussed above, in a repeated SB experiment ~2/3 of the days when Sleeping Beauty wakes up correspond to a result of tails; only ~1/3 of them correspond to a result of heads. Okay, so my point is this: in the original SB experiment, when Sleeping Beauty is awakened and asked “What is your degree of certainty that the coin landed heads?” her situation is the same as that of the mathematician who was called by operator B. That is, each time Sleeping Beauty is awakened during the experiment she knows 1) that the coin has been flipped, 2) that it is Monday or Tuesday, and 3) that she is awake. Therefore she should answer “1/3”. As far as I can see, to take the “halfer” position implies either believing that the mathematician called by operator B should actually say “1/2” (but wouldn’t that be incorrect based on the rules of conditional probability?), or believing that at the moment when the question is posed to her, Sleeping Beauty is in the same situation as the mathematician called by operator A (but she isn’t, because Sleeping Beauty certainly knows that she is awake). So … what am I missing? 😀 • Pace P. Nielsen says: @McLeod I would argue that the question "What is your degree of certainty that the coin landed heads?" could be interpreted in at least the following two ways. #1: "What is your degree of certainty that this is a wakening with a heads flipped?" This appears to be what Pradeep wants in his PIP, and how you are interpreting the question. (But note, this question is asking about more than merely the state of the coin, but also the state of affairs in the experiment.) The answer here is 1/3. #2: "What is your degree of certainty that, at the flipping, a heads was flipped?" Here, it is irrelevant that the scientists are making twice as many measurements in the tail's case. Those measurements have nothing to do with the actual flipping. This is Pradeep's AIP. The answer is 1/2. So, it all ultimately boils down to whether the question being asked refers to the coin's state alone, or to the state of affairs that SB finds herself in and the coin's relation to that state. • Pradeep Mutalik says: @Michael, I apologize – I did not specify the exact conditions of the Wednesday bet I wanted you to make. Here they are: The SB experiment takes place as described before. On Wednesday, SB is awakened by the experimenter and told, "It is Wednesday, and the experiment is over. However, out of curiosity, we are going to ask you the question once more: what is your degree of certainty that the coin came up heads?" The experimental details remain the same, we just want to know if her degree of certainty of 1/3 persists in time or not. The answer is that 1/3 is not the correct answer anymore but 1/2 is (as in your second modification). This was also the case on Sunday. How do we make sense of this? 1/3 is the answer to the question "What is the probability of heads of the coin SB would encounter in the safe on Monday/Tuesday" (the PIP, or the "encountered" proposition). 1/2 is the answer to the question: "What is the probability that heads came up when the coin was tossed on Sunday" (the AIP, or the "tossed" proposition). Note that this latter answer does not have to be half: it could have been different if the SB had some real information about how the coin landed – for example, if her frequency of being awakened on were based asymmetrically on how the coin actually landed. But that is not the case here: the experimental conditions that produce the 1/3 answer for the "encountered" proposition were decided beforehand, and happen the same way every time, regardless of how the coin landed on the toss. So if SB interprets the question as relating to the 'tossed' proposition, her answer would be 1/2, even on Monday and Tuesday, because if she has no information about this on Wednesday, she certainly does not have it on Monday and Tuesday. If SB interprets the question as relating to the 'encountered' proposition her answer as you rightly state, is 1/3. See if SB's own explanation in the dialogue with the EA in the main article makes sense now 🙂 I know that people prefer one interpretation or the other, but linguistically both propositions are valid interpretations of the problem question as I explained. If I still can't make you see the other side, then it means I am a lousy explainer, or that the degree of entrenchment that this problem creates is just too…yuge 🙂 • Pradeep Mutalik says: @Pace, Re. your answer to Macleod. Bingo! That's exactly what I've been trying to say throughout. If you can sell this to the many holdouts still here, you're a better explainer than I am 🙂 • Pace P. Nielsen says: @Pradeep, I'm glad I finally understood your point-of-view. What confused me originally is that I didn't realize when you were talking about the property the coin has (of heads or tails), you were also implicitly emphasizing the time at which this property was to be assessed. To put it another way, when you wrote "What is your degree of belief that the preserved coin in the safe is showing heads now?", this confused me initially because I didn't realize that the word "now" is central to the question. You are asking SB not only about whether the coin has heads in the safe, but also about that property in relation to the fact she is awake *now*. • Pradeep Mutalik says: @Pace Thanks for the feedback. I was trying to express the question as it would be put to SB when she was awakened. Maybe I should have added "now that you are awake". I've been looking for ways to make the concept clearly. What do you think of the tossed/encountered distinction in my "Really simple statement of the Groisman/Mutalik argument" April 12 at 10:57 pm above? Do you think that's clearer? I do want to address your betting comment later, when I have some time. Much of it is covered, though, in point 8 of my formal argument post, followed by my reply to eJ on April 10, 2:26 am (starting about paragraph 4 – the de Finetti procedure…) and some of the back and forth between us later. • Alan White says: Pradeep (if I may) you have certainly illuminated facets of SB I'd not considered–thanks for your labor on this. Pace (if I may)–your remarks at 4:33 and 5:57 are *particularly* sharp and focused–I appreciate them in relation to Pradeep's points. • Alan Cooper says: The question of "how confident" you are that something is true is about what kind of risk you are prepared to take by acting as if it is true in order to achieve some specified reward if it actually is. And this usually corresponds to the relative frequencies which we would expect to see if the whole game is repeated. And so, as with all probability problems, it depends on what the "game" is. Your description of the two approaches may just be a rather convoluted way of distinguishing between the games where you are asked to pay some amount for the promise that you will 1) pay for the game each time you are awakened and get a dollar put in your bank account every time you give a correct answer of "Heads" or 2) only pay once before going to sleep and get one dollar put in your account on Wednesday if the coin is a Head. In other words, if the bet has already been made you can say you always expect that you will win half of the time but if making a separate bet each day you will win by saying Heads only 1/3 of the time. To my mind, the latter (ie game 1) is a more natural interpretation of the question "how confident are you that the coin now shows a head" since that seems to ask for a new commitment on each awakening rather than just the re-assertion of a previously made risk assessment. • A. McLeod says: @Pace, First of all, thanks very much for your response to my comment! I'm still not convinced that the original SB problem can be understood to be equivalent to Pradeep's AIP. I'll try to explain why. First, just to draw a contrast, let's suppose that Prince Charming knows about this experiment too, but he's not allowed any contact with Sleeping Beauty or told anything about the state of the coin or SB's awakenings until Wednesday. So, the experiment is running as originally described, and Sleeping Beauty is awakened by a researcher. (From Sleeping Beauty's point of view right now, it could be either Monday or Tuesday.) The researcher flips a fair coin and hides the result from Sleeping Beauty, and then asks her "What is your degree of certainty that the coin I flipped just now landed heads?" Sleeping Beauty should respond "1/2" (obviously). Then the researcher says, "So, about that coin I flipped on Sunday. What is Prince Charming's degree of certainty that it landed heads?" Sleeping Beauty should once again respond "1/2" (obviously, I think). Then the researcher says, "Once again referring to the coin I flipped on Sunday: What is YOUR degree of certainty that it landed heads (given that you are Sleeping Beauty, you know how this experiment is set up, and you know that you have just awakened)?" And this is where things may start to get contentious, but I think that if the researcher asks the question in that way, including the parenthetical, then Sleeping Beauty is in the same situation as the mathematician who was called by operator B in my previous comment: she is assessing the conditional probability that the coin landed heads, given that she is awake to be asked the question. It's equivalent to Pace's #1 @4:33pm. Sleeping Beauty should respond "1/3". Right? But if the researcher asks Sleeping Beauty the last question without including the parenthetical — if the researcher just asks Sleeping Beauty "About that coin I flipped on Sunday: What is YOUR degree of certainty that it landed heads?" as in the original formulation — I do not think that she can possibly ignore the fact that she does know that she is Sleeping Beauty, and she does know that she has been awakened, and therefore that she is assessing the conditional probability that the coin landed heads given that she has been awakened. Again, let me emphasize: when Sleeping Beauty is awakened during the experiment, she knows that for any coin unconnected with the experiment the probability of heads is 1/2, and she knows that Prince Charming's degree of certainty that the experimental coin landed heads is 1/2, but her OWN degree of certainty that the experimental coin landed heads is 1/3, because she has information that Prince Charming does not have: she knows that she just woke up. That's what I meant at the start of my first comment when I said that I don’t see how one could take the "halfer" position without ignoring part of the available information in the problem. The question is being asked of Sleeping Beauty, and she knows that she is awake. I don't see how she can discard that information when she calculates her own degree of certainty. • Pradeep Mutalik says: @rjt I apologize and retract my answer – I had completely misunderstood your time-reversal argument. I had assumed that the interview was going to be held on Wednesday. It was only after reading Michael's reply to you that I realized that the interviews would be on Monday and Tuesday and that SB would be asked "What is your degree of certainty that the coin will be placed heads up on Wednesday?" I agree with Michael: my answer will be 1/3 on Monday and Tuesday. It would be 1/2 if I were asked on Sunday at the start of the experiment or on Wednesday after the end of the experiment. I now appreciate how brilliant your time reversal construction is! In your version, Groisman's two propositions become: 1) What is your degree of certainty that the coin will be heads up under the setup of being placed? 2) What is the degree of certainty that the coin will be heads up under the setup of awakening? I agree that people may interpret the interview question in either of the above ways. People who are pragmatic and want to be right as often as possible will be thirders and consider proposition 2. People to whom some kind of Platonic truth is important may consider proposition 1. They may reason that the answer was 1/2 on Sunday and also will be 1/2 on Wednesday, and by continuity and the fact that no information can be gained about the future, the "true" answer, even on Monday and Tuesday should be 1/2. I agree that you have proved your point. You have shown that the ambiguity is built into the very structure of the problem, and that the past tense issue is just an interesting curiosity, but is not critical to the ambiguity. I congratulate you, and I agree that this does lessen my contribution to the problem. • Pradeep Mutalik says: @rjt Actually, I take back the last part of what I said. I do congratulate you of course, but the problem still arises why we are able to construct two meanings out of one statement, and my point about natural language still applies. I had said "These two meanings arise in human speech and writing whenever anything is expressed about a different time, and specifically whenever the past tense is used." This also applies to the future tense. When you are asked, "What is your degree of certainty that the coin will be placed heads up on Wednesday?", you still have to decide where in time you place yourself. Do you place yourself in the current time, or do you place yourself on Wednesday? If you do the first and place yourself today, you are thirder. If you do the second and imagine yourself on Wednesday in tune with the rest of the sentence, then you are a halfer. So the problem is perfectly symmetrical in time, and the same problem arises for the future tense as it does for the past tense. In everyday communication, the context of the utterance will help you to decide this. But in the statement about the coin, both meanings are up for grabs. So thirders are firmly fixed in their own time, halfers are time travelers, whether it is the past or the future! 🙂 • Pradeep Mutalik says: @eJ You said: Groisman appears to be a Thirder: "I agree with thirders that her credence on awakening should be 1/3". No, Groisman is a unifier. He also agrees with halfers that the credence under the setup of coin-tossing should be 1/2. This is clear if you understand what the two propositions are. Under Groisman's explanation, the single statement about credence is transformed into two statements about actual probabilities. The "credence under the setup of awakening" is my PIP or "encountered" proposition. Under this interpretation, SB is being asked on being awakened, "What is the probability that the coin in the safe is showing heads now?" Not credence, actual probability. The kind you can confirm by doing the test many times. There is no way you cannot be a thirder about this. Just as there is no way you cannot be a halfer about the other proposition: the probability of getting heads on the coin toss. That is why, in the PIP or "encountered" version of the extreme Sleeping Beauty variant (where she is awakened a million times on tails) the answer is 1/1,000,0001. This is her chance of actually encountering a coin showing heads in the safe. Of course, her answer to the AIP or "tossed" proposition is exactly one half, and there is no conflict whatsoever between these two numbers. To give different answers to the two propositions is to be plain wrong, because they are objective problems in probability, with the subjective aspect being surgically taken out. Mix up the two and you get chaos. Which reflects much of the thinking on this fascinating problem. • rjt says: @Pradeep, Ah, I am glad that you have understood and agreed with my reasoning. Sorry it was not clear beforehand. Now it is my turn to be confused. You ask "Do you place yourself in the current time, or do you place yourself on Wednesday?" This works perfectly in the original SB problem. Either you're considering the coin right after the moment of flipping, or you're considering the coin right now. However, I can't grasp how it works in the inverted-time version. If you place yourself on Wednesday, great, you're considering the probability that the coin will be placed on one side or another. But if you place yourself in the current time, the coin has *not yet been placed*. The question is unanswerable. It's like asking someone with no children the possibility that they have a boy or a girl. It's 0 in both cases. The question is ill-formed. Still, as @Michael observed, the thirder position is still tenable in the inverted-time version. Taken together, it seems to me that this points toward the ambiguity not being related to temporal reference of any sort. I'd like to suggest, then, that the ambiguity really has to do with how people interpret the deontology of SB's response. There are two camps here: A) She should attempt to be correct in the maximum number of *cases* (distinct events where she has an opportunity to bet). B) She should attempt to be correct in the maximum number of *worlds* (distinct possible universes). A is the thirder interpretation, one that is easily illustrated through consideration of optimal betting strategies, as many commenters have shown. B is the halfer interpretation, one that is easily illustrated through consideration of, well, the a priori probability of coin tossing. What makes the SB problem different from most other probability problems is that usually the number of *worlds* and the number of *cases* are exactly equivalent. Regardless of whether you're trying to optimize a blackjack strategy that works in all possible worlds or in all possible cases, you'll get the same result. This is why this issue of interpretation never arises. The SB problem, however, has a mismatch. If you are optimizing for maximal correctness over all possible worlds, you will have a strategy (the halfer strategy) which is quite different from optimizing for maximal correctness over all possible cases. • Pradeep Mutalik says: @rjt Just briefly how it works in the inverted-time version is that you place yourself in the current time, and maximize your chance of being right. • Pradeep Mutalik says: @eJ PS to last comment Before you protest that removing the subjectivity makes it a different problem, let me hasten to add that the subjectivity is just brought up front. SB has the subjective freedom to choose which proposition she wants the question to mean. Once she makes that choice, everything is deterministic. She has thrown the switch on the rails and has to follow wherever that branch of the track leads: to halfer land or thirder land, as the case may be. • Ted says: Perhaps this is naive, but this problem seems analogous to a very different problem: Evaluate the series 1-1+1-1+… Bracketing terms as so gives: (1-1)+(1-1)+…=0+0+…=0, or 1+(-1+1)+…=1+0+0…=1. We have two contradicting answers, that both feel correct with no obvious flaw in the method i.e. the same issue that has caused the debate. We can debate whether one persons method is right or not, but clearly this is due to the fact that the sequence 1,-1,1…does not converge, the problem is ill-defined. Perhaps it would be possible to reduce the sleeping beauty problem to this? Interestingly we can use analysis/calculus to get another answer of 1/2, perhaps there is a compromise to the sleeping beauty problem? • Pradeep Mutalik says: @rjt You are right about the ambiguity of cases vs worlds or wakings vs experiments. This ambiguity was discussed at length in our previous discussion of the problem in my Quanta Insights column back in January. The insight I had subsequently was that this ambiguity is backed up by the linguistic temporal ambiguity which supports both the options and therefore leaves both camps with the feeling that they have answered the problem. This linguistic ambiguity also prevents people from immediately pointing out "You have misinterpreted the problem statement" which you can say in other problems. Both camps can claim that that they have addressed the problem statement. Thus the cases vs worlds ambiguity and the language temporal ambiguity dovetail each other. • Michael J Ahles says: There is no certainty in probability so the answer is zero. = • eJ says: Pradeep, Groisman is no unifier. He says that 1/2 and 1/3 are correct answers to two different questions, but that only the latter is the correct one. He doesn't give SB permission, as you do, to interpret the question either way. He's a Thirder. Your claim that "SB has the subjective freedom to choose which proposition she wants the question to mean" is hopeless. You're asking her to comment on two Boolean propositions, i.e. (P1) coin-landing-Heads and (P2) Heads-in-a-safe, that are logically identical. Check the four cases! — it's Heads in P1 iff it's Heads in P2. So, being a rational agent, her credence for P1 is exactly her credence for P2. You can't believe something 50% and simultaneously believe exactly the same thing 33%. Imagine scanning her brain when telling her that if it's Heads she'll get a tasty treat: as a rational agent, the "reward neurons" must "light up" (sorry, I'm no medic) to exactly the same level of excitement whether we give her the news P1-style or P2-style, because (let me repeat) P1 and P2 are logically the same thing. So linguistic ambiguity has no place in SB's thinking about the interview question. But it does have a significant role in *our* thinking about the *puzzle*, because it can lead people into modelling the wake-ups one way or the other. It's perfectly OK for someone pondering SB to be tricked into Halfism by the coin-landing-Heads or Thirdism by the coin-in-a-safe, but then what should they say when presented with the other phrasing of the problem? The respectable positions are to either commit to a single camp, Halfer or Thirder, or to declare that they are confused and need more time to think. It is a nonsense to project that confusion onto SB and declare her a Dualist that answers differently in the two cases. That's what I think you're missing about Groisman and SB linguistics more generally: it offers an excuse for the confused puzzle reader, but says nothing about the deeper question of reconciling committed Halfers and Thirders who are unmoved by P1-vs-P2 rephrasing of the question. • Murray says: Way too many words, there is no "time warp", stop wasting time thinking from the experimenters perspective, SB does not mess with any saved coin toss locked away, and there is no gambling in this game! This is simple, just answer the question as if you were SB figuring this out before going to sleep or upon any of her independent awakenings. We all agree that each of the three M, T awakening states occur with equal probability (H,M), (T,M), (T,T). If SB knew it were M, then she would answer .5 and if she knew it were T, then she would answer 0, but she does not know the day. A weighted average of the right answer over those three states is 1/3. She chooses 1/3. Simple. The Tails coin toss enables SB to vote twice when Tails occurs, whereas she can only vote once if Heads occurs, this fact is what confuses folks. The infinite Tuesdays folks are getting at this above, with Tom nailing it with more words. Some of us wish: Please vote for Hillary or Bernie, but if you vote for Bernie you can vote twice – but did Bernie really win if only 1/2 the voters supported him?! Ask the eight member Supreme Court! • Pradeep Mutalik says: @eJ I apologize, eJ. From your comment it is clear that I have done a terrible job at conveying the essence of my argument. Here’s how P1 and P2 are completely different. P1 (coin landing a particular way) is a single Boolean event: heads or tails with probability 1/2. P2 (coin showing heads or tails in safe, being encountered by SB on an awakening) is a multitude of events that can be manipulated independently from P1 by changing the experimental conditions. These encounters can be made asymmetric between heads and tails, by “hiding” the heads —by not having some encounters where the coin would have shown heads. In the extreme SB problem there are 1,000,001 events, only 1 of which is a heads encounter. In the standard SB problem there are 3 events, only 1 of which is a heads encounter. It’s the probability of a heads encounter relative to all encounters that we are estimating in P2. So P1 and P2 are not the same at all. Depending on which of P1 or P2 is chosen as the question being asked to SB, the answers are different. But because these two propositions are independent, we (and SB) are free to choose both: we can be halfers about P1 and thirders about P2. There is no conflict between the two. I thought this was pretty clear in the SB interview: “Let’s return to your question, which is actually two separate questions. The first question is: What is my degree of belief that the coin landed heads in the act of being tossed? This value, of course, was one-half on Sunday, and will remain one-half until I actually find out what happened. I am a true halfer about this. The second question — what is my degree of belief that the preserved coin is showing heads now — is the same as saying, “What is my degree of belief that the coin in the safe landed heads sometime in the past?” On Monday and Tuesday, it is one-third, because I am in a time warp with half the heads lost in time. I am definitely a thirder in response to this question when I am awakened on these two days.” Perhaps the time-warp metaphor threw you off. If so, I’m sorry about that. I’ve tried many explanatory devices to make the argument understandable. And I’ve also explained it more plainly elsewhere. I’ll keep trying, if you ask for specific clarifications. Talking about committed Halfers, Pace P. Nielsen was clearly one, both in the January discussion and earlier here. But now he has clearly understood and expressed the argument in a way that has been praised by others. Maybe you’d like to check out his last two comments. Let’s drop Groisman for a while. Maybe our understanding of him is different. I’ll read his paper again. • Brett says: This proposal doesn't seem like it's going to work. Let's start from the beginning: we're considering a version of the Sleeping Beauty problem in which the coin, after it's tossed, is preserved in a safe in the state in which it landed. So, when Sleeping Beauty is woken up and asked the question by the experimenter, she has to choose between two propositions to evaluate. Here they are: P1: The coin landed heads in the act of being tossed. P2: The preserved coin is showing heads now. It's true that these are distinct propositions, but it's just not possible for an ambiguity between them to motivate two different answers to the experimenter's question. The problem is that, on this setup, Sleeping Beauty knows that the coin is preserved in whatever position it initially landed in. That is, she's certain that P1 and P2 have the same truth value. So if C(*) is the probability function that models her belief state, then C(P1 if and only if P2) = 1. Now, if Seeping Beauty is perfectly rational–as we're assuming she is–then her beliefs are going to be probabilistically coherent. That is, C(*) will obey the rules of probability theory. And it's just a theorem of probability theory that, if C(P1 if and only if P2) = 1, then C(P1) = C(P2). So Sleeping Beauty has got to have the same degree of confidence in P1 that she has in P2, which means that, whether she interprets the experiementer's question as asking about P1 or as asking about P2, she's going to give the same answer. THere's no way that halfers and thirders can both be right. • Pace P. Nielsen says: @McLeod, I agree with quite a bit of what you said. I think the crux of the problem comes when you say: "But if the researcher asks Sleeping Beauty the last question without including the parenthetical — if the researcher just asks Sleeping Beauty "About that coin I flipped on Sunday: What is YOUR degree of certainty that it landed heads?" as in the original formulation — I do not think that she can possibly ignore the fact that she does know that she is Sleeping Beauty, and she does know that she has been awakened, and therefore that she is assessing the conditional probability that the coin landed heads given that she has been awakened." My response would be that she can indeed ignore the fact she knows about the experiment, but only if she believes that the question they are asking her is implicitly telling her to do so. To put it another way, you gave multiple ways of expanding the question (or adding "parentheticals"). Your position seems to be that if the scientists do not explicitly clarify which expansion they are talking about, then SB should assume they are talking about her state of belief in reference to the awakenings. I would actually agree that this is likely, for otherwise why go to the trouble of creating such a convoluted experiment in the first place? But then again, maybe they are testing her ability to ignore irrelevant information, so when they are asking the question they want to see if she can separate the probability of the coin flip from the possible states in the experiment. So, ignoring information may not be a problem after all. (Math teachers do it all the time with word problems.) SB can certainly disregard information if she believes that the scientist's question is implicitly asking her to do so. On the other hand, she cannot disregard information if she believes it is crucial to the scientist's question. And, in my opinion, she does not necessarily have enough information to decide which of these two cases actually holds, so she should ask a clarifying question. • Brett says: I just looked back, and it turns out eJ makes essentially this point. I apologize–I should have read all the previous comments before commenting. Now that I know, though, it makes sense to say something about Pradeep's response to eJ. So, here goes: Pradeep says the following: "P1 (coin landing a particular way) is a single Boolean event: heads or tails with probability 1/2. P2 (coin showing heads or tails in safe, being encountered by SB on an awakening) is a multitude of events that can be manipulated independently from P1 by changing the experimental conditions." This seems confused. Neither P1 nor P2 is an event (or a multitude of events). They're propositions. (Here's an easy way to see that a proposition can't be an event: propositions have truth values, but events do not.) Now, here's another way to see that Sleeping Beauty cannot have a different degree of confidence in P1 than she has in P2. Imagine the experimenter wakes her up and asks the following questions: 1. How confident are you that the coin landed heads at the time of tossing but now, preserved in the safe, does not show heads? 2. How confident are you that the coin now, preserved in the safe, shows heads, but it didn't land heads at the time of tossing? If Sleeping Beauty is rational, she should think that (as long as the experimenters haven't lied to her about what they did with the coin after they tossed it) each of these scenarios is IMPOSSIBLE. And that entails that she should be certain that (P1 if and only if P2) is true. The rest of the argument goes as I said in the previous comment: if she's certain that (P1 if and only if P2) is true, she can't rationally have a different degree of belief in P1 than she has in P2. • eJ says: No, Pradeep. Just no. At the time of asking, P1 and P2 are Boolean-valued: they are both true or both false: the coin *has* landed and *is* showing whatever it's showing. They are identical logical propositions, so SB's credence in them must be the same. If we can't agree on this then we should just congratulate ourselves on having identified yet another way that the puzzle can divide people. That's it. • Pradeep Mutalik says: @Brett (and eJ) Yes, I accept that I was loose with my language there – I just scribbled that comment hurriedly before rushing off to catch a plane. Let me try it again. The two propositions are: P1 = the coin landed heads in the act of tossing. P2 = the coin shows heads now (when SB is awakened). It is true that when P1 is true, P2 is also true. As you rightly said, the opposite result is impossible. Now let us determine the probabilities of P1 and P2 being true. The probability of P1 being true is 1/2. No argument there. SB is asked the truth of P2 multiple times. For every 3 times that SB is asked the truth of P2, ~P1 is true 2 of the times (the coin has landed tails on tossing) so P2 is false twice. So the probability of P2 being true is 1/3. Hence SB's credence in P2 should be 1/3. Notice that the fact that P1 implies P2 has no bearing on the probability of P2 being true, because we can group one instance where P2 is true with multiple instances where P2 is false. In fact, in general we can manipulate the probability of P2 being true to any rational value whatsoever within the closed interval [0, 1] by suitably changing the experimental conditions and grouping appropriate numbers of true and false P2s. We can do this without ever changing the probability of P1 from its value of 1/2. Therefore the probabilities of P1 and P2 are independent. You can be a halfer for P1 and a thirder for P2. Those are, in fact, the right answers for the SB problem. • eJ says: You're on repeat, Pradeep, and nothing in your flat-wrong argument about SB's credence for P1 and P2 being different gets any better over time. SB's credence will be with respect to whatever reference frame / sample space she deems correct. Halfers and Thirders have different views as to what this should be, but Beauty, being fully rational, must pick just one. *She* must thus have one credence for Heads, even if *we* may be prone to leaning one way or the other based on the way the puzzle is phrased. The Dualist position is irrational and not "in fact, the right answers for the SB problem". How much more of Quanta's bandwidth do we need to waste talking across each other like this? • Pradeep Mutalik says: @eJ, Why is it irrational to have two different credences about two different, independent propositions? • Pradeep Mutalik says: @eJ, …or to be more accurate, why is it irrational to have two different credences for two different propositions whose probabilities can vary independently? (Thanks for keeping me honest and for keeping my language tight, guys 😀) • Brett says: It's just a basic fact about probability: if Pr(*) is a probability function, then if P implies Q, then Pr(Q) is at least as high as Pr(P). (The intuitive reason for this is that every possible case where P is true is ALSO a case where Q is true, and for Pr(Q) to be lower than Pr(P), there must be cases where P is true and Q is false.) Now, we agree that P1 implies P2 and vice versa. (This means that, while P1 and P2 are different propositions, they certainly are NOT independent of each other.) So, Pr(P2) is at least as high as Pr(P1) and vice versa, which means that Pr(P1) = Pr(P2). If Sleeping Beauty is perfectly rational, her degrees of belief are probabilistically coherent; i.e., C(*) is a probability function. So Cr(P1) = C(P2). (Note that the number of times she's asked the question isn't relevant to anything I've said here. I've relied only on a basic fact about probability theory and on the fact that ideally rational credences are probabilistically coherent.) The problem with your proposal is that it's inconsistent with basic probability theory. • Pradeep Mutalik says: @Brett, Your first paragraph does not apply if propositions P and Q are about items that belong to different sample spaces. Then the sample space of Q can have extra elements that can “dilute” the probability without needing to have cases where P is true and Q is false. This can make Pr(Q) lower than Pr(P). For example, let set A={1,2} and set B={1,-1,2}. Items are picked from set A and set B according to the following rules: If 2 is picked from set A, then 2 is also picked from set B. If 1 is picked from set A, then both 1 and -1 are picked from set B. Proposition P is “2 is picked from Set A.” Proposition Q is “2 is picked from Set B.” Obviously, P implies Q. But Pr(P) among items picked form Set A is ½ and Pr(Q) among items picked from set B is 1/3. You can see the parallel with the SB problem. The sample space for P1 is {Heads on toss, Tails on toss}. But the sample space for P2 is {Heads in safe Monday, Tails in safe Monday, Tails in safe Tuesday}. The Pr(P1} is ½, but Pr(P2) is 1/3. So my proposal is perfectly consistent with probability theory and your objection does not apply. • Brett says: No. A sample space is not just a set of possibilities. It's a set of EQUALLY LIKELY possibilities. To appeal to facts about sample spaces, you need to have already decided which possibilities are equally likely, which is precisely what's at issue here. You can't just read the answer off the language in which the question is asked. (Indeed, one way of describing the disagreement between halfer and thirders is the following: halfers think the sample space is {The coin landed heads, The coin landed tails}, and thirders think the sample space is {The coin landed heads and it's Monday, The coin landed tails and it's Monday, That coin landed tails and it's Tuesday}.) To put the point another way: propositions are NOT, in any relevant sense, "about items that belong to…sample spaces". They're just about what the world is like. For instance, the proposition that the coin is showing heads now, in the safe, tells you nothing, on its own, about what sample space you should use for evaluating it. As for your example: if the rules say to pick at random from set A, you're right that Pr(P)= 1/2. But, assuming we're evaluating P and Q after a single trial of the experiment, you're just wrong about Pr(Q)–it's 1/2. Given the setup of the experiment, the relevant sample space is not {2, -1, 1}–it's {2, (-1 or 1)}. And this is precisely BECAUSE what happens with set B is fully determined by what happens with set A. Note, too, that your example isn't precisely analogous to the Sleeping Beauty problem. (I'm not saying that you think it is–I'm just trying to make sure this is all as clear as possible.) The question of whether Q is true, asked after a trial of the experiment, is analogous to the question of whether Sleeping Beauty was woken up only once, asked after the experiment is over. And everyone should agree that, after waking up on Wednesday, when the experiment is over, Sleeping Beauty's credence that she was woken up only once during the experiment should be 1/2. • Peter says: SB knows for a fact the coin was only flipped once & it is a fact that she was only asked her belief [if] the coin landed heads? [after one flip]. SB was not asked if the coin currently reflects the original outcome, this is irrelevant as seeing the coin during the interview or picturing its current state in a safe is not part of the experiment, if it was it should have been stated, so this should be automatically dismissed. it has to be 50:50. • PY says: I wonder if the problem written in different languages would elicit different distributions of respondent types? • Pradeep Mutalik says: @PY, That's an interesting question. Linguists I spoke to, however, are skeptical. All human natural languages have evolved to make it easier for us to express what we mean without having to specify everything down to the last detail. When we refer to a different time or place, many questions arise for the listener: should the listener situate herself in the time and place where the conversation is taking place, or should she situate herself in the time and place that the conversation is referring to? If you had to specify that every time, language communication would be tedious. So we rely on context that enables us to understand the speaker's intent, say, 99% of the time. In the Hindi language, for example, the same word means both "yesterday" and "tomorrow"—you have to divine the speaker's intent by context and the tense of the verb. This context dependence is what makes it easy to communicate naturally, and why you can in most cases be quite sloppy in details, say things like "that thing" or "whatchamacallit" or "whatshisname" and still be understood based on context in most cases. This reliance on context makes utterances ambiguous sometimes, which is why natural language utterances sometimes have double meanings, puns, subtext etc. In context-free situations like mathematical problems or computer programming or dealing with computer interfaces, this is a big problem, and that is why intent has to be expressed to the last detail to prevent being misunderstood. It's one reason why people often get exasperated with computers. Of course, every natural language has the tools to express intent precisely in such situations – it just requires a lot more effort from the speaker. In the SB problem for example, the speaker could say, "Think about the coin toss on Sunday. What is the probability that the coin landed heads?" or "Think about each time you examine the coin in the safe on awakening. What is the probability that the coin shows heads?" • Peter says: @Pradeep We are told that the setting is an experiment therefore nothing should be left to assumption and everything should be laid out like a computer program, there is no Safe for the coin in the experiment & we must take everything on face value, we can't say what if it the question was asked differently that would make the experiment a different experiment. • Pradeep Mutalik says: @Brett, I agree with two things that you said. You are right that "…one way of describing the disagreement between halfer and thirders is the following: halfers think the sample space is {The coin landed heads, The coin landed tails}, and thirders think the sample space is {The coin landed heads and it's Monday, The coin landed tails and it's Monday, That coin landed tails and it's Tuesday}. You are also right that on Wednesday, after she is told that the experiment is over, but before she is told how the coin actually landed, SB's credence for heads should be 1/2. In my proposal, I have shown that the problem statement can be understood in two valid ways. The two sample spaces you described apply to the two ways of understanding the problem. That is why it is possible to assign the halfer solution to the first, and the thirder solution to the second with no conflict. On Wednesday, the two interpretations coincidence, and both give the answer 1/2. All this is described in the article and in my many comments. My proposal, once you understood it correctly, resolves all the paradoxes of the SB problem. Your objection stating that if a proposition P implied proposition Q then their probabilities will be identical, was interesting. I showed how that can come about and gave a simple counter example. It seems to me that all your objections since then are in the nature of 'reaching' – you seem to be desperately searching to try and find something wrong with my solution. First, your statement about sample spaces having to have equally likely possibilities is irrelevant for the SB problem and wrong in the general case. Both the sample spaces you described for the SB problem do have equally likely possibilities. Secondly a sample space is defined as the set of all possible outcomes of an experiment, each one of which should have a positive probability measure (not necessarily equal) with the sum of all the measures equalling up to 1. I looked up a bunch of these definitions and not one insisted that the measures have to be equal. Anyone who has done real world probabilities knows that the kind of ideal space you are describing is a pipe dream – yet you can apply probability theory without a problem. Your second objection concerned my counter example. You stated "But, assuming we're evaluating P and Q after a single trial of the experiment…". This is pathetic. Probability is not about running single trials. It is about counting all the positive outcomes and dividing by the number of all possible outcomes. In both my counter example and in the SB problem, you need at least two trials to completely cover all outcomes and the result, for both propositions Q and P2 is 1/3. Third, you said that propositions are about what the world is like. Yes, but they could also be about what the world is like at a specific time and place as is the coin in the safe proposition. The sample space you use to evaluate the probability of this is dictated by the question you are trying to answer. In my second interpretation of the SB question, it is clear the you have to evaluate it across all wakings, because the question is asked at all wakings. Considering how much you seem to be reaching to raise weak objections, I have to ask you your motivation for not accepting, or at least deeply considering my proposal. Are you just repeating conventional talking points about the SB problem? Do you not accept the thirder position that the probability of HM, TM, and TT are 1/3 each? Then you should state specifically what objections you have to the arguments given here by people like Rich, Michael, Murray and others who have shown this. Is it something else? I look forward to hearing what your motivation is for rejecting this solution really is. If you do have an interesting reason I'll be glad to respond. • eJ says: What on earth?! Pradeep, pull yourself together. Brett has argued in a pretty even-handed way, and now here you are calling his arguments "pathetic", "reaching to raise weak objections" and questioning his motivation?! You're in a privileged position as article author, representing a magazine whose mission is to enhance the public understanding of science. I was appalled enough previously that you were stating as fact that your take on SB was correct (is that how science works?), but hurling insults at the readers who gamely partake in your request to critique the article? … very poor. Very poor indeed. • Peter says: There are only two possible outcomes SB either slept through Tuesday or she was awoken on Tuesday. • Pradeep Mutalik says: @eJ, I've spent hours patiently answering questions and objections, including yours and Brett's, whose original objection I even called interesting. Even now, I welcome objections made in the spirit of open minded inquiry. I clearly listed where I agreed with him. The only time I used the word 'pathetic' was on a beginner level objection that, considering the otherwise high level of knowledge that Brett displays, was clearly something he knows better about. In a public forum like the comments section here, it is an unfortunate fact that we sometimes encounter people whose sole purpose is to be naysayers. Most authors don't even bother to try and answer every one's comments. But you are right, I should not insist that mine is the only correct interpretation. That's an operational hazard for anyone who has an original idea, and to some extent it is necessary to put an idea forward. Nevertheless, I apologize for succumbing to it here. I sincerely apologize to anyone who I might have offended, and specifically Brett and yourself. Please keep your genuine questions coming. I will do my best to answer them honestly, patiently and in good humor. 😀 @Brett, I sincerely apologize to you for using strong words in the heat of battle. That's what the SB problem does. And that's what makes it fascinating. It's a nice feeling we can share. 😀 • Brett says: Re: sample spaces. You're right that, technically speaking, it's possible for a sample space to be such that the possibilities aren't equally likely. What I should have said is that, IF you want to make use of sample spaces in calculating probabilities in the particular way you do above, it must be a UNIFORM sample pace–i.e., the sample space must be set up in such a way that the possibilities are equally likely. Here's why: you infer directly from the fact that there are three possibilities in the sample space to the claim that the probability of each possibility is 1/3. Obviously, this is not a valid inference if the three possibilities are not equally likely. (If that's not the inference you're making, let me know. But it certainly seems it is, both in your 4/16 2:17PM comment and in your most recent comment, where you say that probability is about "counting all the positive outcomes and dividing by the number of ALL possible outcomes".) I did make a mistake here with my language, though, and for that I apologize. I considered submitting another comment to correct myself, but I didn't want to spam the comment section. On to the rest of your response. First, I ask you to refrain from psychoanalyzing me. (Thanks for the apology, by the way. I started composing this response before you posted it.) For what it's worth, I'm not a halfer–in fact, I'm attracted to thirderism—but that's beside the point. I don't have a desire to defend any particular position here. My only motivation, since you're interested, is this: it seems very clear to me that your proposal is, for very basic reasons, probabilistically incoherent, and I hope that I can convince you of this so that you will pass this information along to your readers. (I apologize for the directness here, but you asked the question.) As for my objection, it remains the same as it has been from the beginning: if P and Q imply each other, then for any probability function Pr(*), Pr(P) = Pr(Q). I've tried various ways of explaining why this is the case, but the fact is this: it's just a (borderline-trivial) theorem of probability theory. I defy you to find any textbook in probability theory that says otherwise. (Notice that this theorem is a simple conditional. It's not subject to any constraints about what sample spaces are in play, etc.) Now, you say that you've given a counterexample to this claim. So, given that the claim is a basic theorem of probability theory, there are two alternatives here: you've made a mistake somewhere, or probability theory is fundamentally misguided. So let's work through the example in detail, to see which one is the case. The setup is this: A member of set A–i.e., {1, 2}–is chosen at random. Then, depending on what's chosen, something happens with set B–i.e., {-1, 1, 2}. If the number chosen from A is 2, then 2 is chosen from B as well, and if the number chosen from A is 1, then both -1 and 1 are chosen from B. Now, here are the two propositions under evaluation: P: 2 is picked from set A. Q: 2 is picked from set B. Let Pr(*) be a probability function–let's say, for concreteness, that it's the probability function that models the perfectly rational credence for me to have at some particular time. Then the claim is that P and Q imply each other, but it's not the case that Pr(P) = Pr(Q). I want to start by noting that each of these propositions is ambiguous. I interpreted them both as being about a particular trial. Here's another interpretation: they may be about what has happened at at least once, during ANY trial of the experiment run so far. But note that, on the second interpretation, both Pr(P) and Pr(Q) go up as the number of trials increases, approaching 1 as the number of trials approaches infinity. In fact, they go up TOGETHER–no matter how many times the experiment has been run, my confidence that the number 2 has been chosen from set A at least once in all those trials should be equal to my confidence that the number 2 has been chosen from set B at least once in all those trials. I assume that we agree here and that this isn't the interpretation you have in mind. Let me know if that assumption is mistaken. But if we go with the first interpretation and interpret both P and Q as being about a particular trial, the reasoning I gave in my previous comment applies: my confidence that 2 is picked from set A in THAT trial should be 1/2, but so should my confidence that 2 is picked from set B in THAT trial. I assume you agree here as well, but if not, you can convince yourself of it, if you like, by running a computer simulation. I assure you that, in approximately half of all trials, it will be the case that 2 is picked from set B in that trial. (This is not "pathetic". It's just accurate.) Again, I'm assuming now that neither of these is what you have in mind, since it's clear in both cases that Pr(B) = 1/2. So I'm guessing you have in mind something like the following: we run a bunch of trials of the experiment, generating a whole bunch of instances of picking from B, and then we pick one of these pickings at random. Q, then, is the proposition that 2, rather than 1 or -1, is chosen in THIS instance of picking from B. Is that right? If so, this much is right: in the long run, approximately 1/3 of the pickings from B are pickings of 2, and so, given that we've picked one of those pickings at random, my confidence that that it's a picking of 2 should be 1/3. So Pr(Q) = 1/3. But notice the following: what you were trying to generate is a counterexample to the claim that if P and Q imply each other, Pr(P) = Pr(Q). But now, what is P? Here's one possibility: if we analogously pick an instance of picking from A at random and say that P is the proposition that 2 is chosen in THIS instance of picking from A, then we do get the result that Pr(P) = 1/2. But in this case P and Q do NOT imply each other–we have no reason to think that the picking from B we've picked at random is from the same trial as the picking from A we've picked at random, and so it's entirely possible that P and Q have different truth values. And since my claim is about propositions that DO imply each other, this isn't a counterexample. Really, the only reasonable way to specify P such that P and Q imply each other is something like the following: P: 2 is chosen from A in whatever instance of picking from A is part of the same trial that the instance of picking from B relevant to Q is part of. That way, it can't be the case, given the experimental setup, that P is true while Q is false or vice versa. But notice that P is no longer about whether 2 is chosen in any PARTICULAR instance of picking from A–the proposition can be made true by different pickings, depending on which picking Q is about. And so, while it's the case that, for any particular picking from A, my degree of confidence that 2 is chosen in that picking should be 1/2, it nevertheless does not need to be the case that Pr(P) = 1/2. So even on this interpretation, there's no counterexample here. It does indeed turn out, since Pr(Q) = 1/3 and P and Q imply each other, that Pr(P) = 1/3 as well. So we're back to the claim that, if P and Q imply each other, then Pr(P) = Pr(Q), which is, as I said, a theorem of probability theory, and to which you have not given a genuine counterexample. And in the Sleeping Beauty problem, it definitely IS the case that P1 and P2 imply each other: as you have already agreed, Sleeping Beauty cannot regard it as possible that one of them is true while the other is false. So Pr(P1) = Pr(P2). This is getting very long now, so I'll stop. I know there are one or two concerns that I didn't address, but I hope this is sufficient for you to see where I'm coming from. • Michael says: @Pradeep Thank you for clarifying the Wednesday bet, as per: "I apologize – I did not specify the exact conditions of the Wednesday bet I wanted you to make. Here they are: The SB experiment takes place as described before. On Wednesday, SB is awakened by the experimenter and told, "It is Wednesday, and the experiment is over. However, out of curiosity, we are going to ask you the question once more: what is your degree of certainty that the coin came up heads?" The experimental details remain the same, we just want to know if her degree of certainty of 1/3 persists in time or not." Yes, I agree that on Wednesday the rational answer is p=1/2 in this scenario. This is because she updates her beliefs given the new data that it is Wednesday. One has the two conditional probabilities p(heads|expt interview) = 1/3 and p(heads | Wed interview ) = 1/2, conditioned on whether SB knows it is the experiment or that it is Wednesday. Any other answers would lead to irrational decisions on bets, as I've pointed out in an earlier comment. What I don't agree with is that the second of these conditional probabilities is a valid answer to the SB problem, as posed at the top of the page. The problem asks for her subjective probability of heads at the time of the experimental interview, not at any other time. She is certainly not asked for p(heads|Wed interview). The idea that probabilities change when conditioned on different data is not new, and not illuminated by the SB problem. A much better problem in this regard is the well known Monty Hall problem, where new data is actually part of the problem. • Pradeep Mutalik says: @Michael Good, we can agree that that p(heads|Sunday) = 1/2, and p(heads | Wed interview ) = 1/2. So we conclude that p(heads|expt interview)=1/3 is caused by the peculiar experimental setup, and does not really tell SB anything about how the coin actually landed in the act of tossing. Now this probability of how the coin actually landed does not have to be 1/2 always – it can change with information. Suppose the assistant tells her that the coin landed heads, but SB feels that she can only trust him 80%. Then her subjective probability for heads will become 4/5. If the professor confirms it, then her subjective probability will become 1. From the fact that p(heads|Sunday) = 1/2, and p(heads | Wed interview ) = 1/2, SB can rightly conclude that in this period, she never received any direct or probabilistic information about how the coin actually landed when tossed. Even on Monday and Tuesday, she cannot have had any information about this. So when SB is asked about the subjective probability that the coin actually landed heads in the act of tossing, she can therefore truthfully answer 1/2 even on Monday and Tuesday. It is a different question from the one about the coin in the safe, which concerns the probability of SB "encountering" heads in the safe, which allows her to make the right bet. The answer to this other question about the coin landing heads in the act of tossing, is just something she can just deduce from the facts above by looking at the state of her knowledge. My point is that there are 2 questions which are both valid interpretations of the problem statement. One has an answer 1/3, and the other has an answer 1/2. The second question may seem to you be a strange way to think about the question, but to someone who is thinking about the truth behind what happened at the coin toss, the betting way seems equally strange. And that's where the conflict lies. Both sides are right in their own way, and each way of thinking seems strange to the other party. This "tossed/encountered distinction" is a subtle point and I didn't explain it very well here. The best way to understand it is to consider SB's numerous examples in the column write-up or the one I gave in my "Easy summary", which I reproduce below. Imagine that 100 fair coins are tossed, and as expected, about 50 come up heads and 50 tails, but subsequently some asymmetric process destroys or hides all the heads and only heads. Now you are certain that the coins were fair when tossed (probability of heads = 1/2), but you can also be certain that any coin you encounter in the future will be one showing tails (probability of heads = 0). The ‘encountered’ value changes depending on what happens after the coin toss, whether some coins were hidden, how often you were exposed to them etc. The ‘tossed’ value only depends on the truth of what happened when the coin was tossed. • Pradeep Mutalik says: @Brett, There is nothing in your last post that disproves my counter-example. The first few paragraphs of your comment contain a lot of irrelevant stuff, and in the last few paragraphs I think you (once again) make the cardinal error of judging probability based on an individual case. Here's a simple illustration of the correctness of my counter example that can be tested on a computer. Here is the counter example again for convenience: Let set A={1,2} and set B={1,-1,2}. Items are picked from set A and set B according to the following rules: If 2 is picked from set A, then 2 is also picked from set B. If 1 is picked from set A, then both 1 and -1 are picked from set B. Proposition P is “2 is picked from Set A.” Proposition Q is “2 is picked from Set B.” Now pick randomly from set A a million times. Call this string of numbers PickA. PickA will be something like 1,2,2,1,1,1,2,2… Generate PickB based on the above rules. There will be about 1.5 million corresponding numbers in it. For the PickA result shown, PickB will start 1,-1,2,2,1,-1,1,-1,1,-1,2,2…. You can put a test in your program to flag an error if (P implies Q) is violated. It will not happen. (Actually you don't really need to do this test, because whenever 2 is in PickA, it is also in PickB, and there are no instances of 2 being in PickB which do not also have 2 in PickA… but satisfy yourself if you need to). Pr(2) in PickA — let's call it x — will be ~1/2. Pr(2) in PickB will be (2/3)*x or ~1/3. Since this shows that my counter example is correct, and we know that the foundations of probability are correct, let me suggest a third alternative. Perhaps you have misstated the theorem you are talking about — perhaps you have omitted its assumptions or the conditions under which it can be applied. So please let us know what this theorem is called, and give a reference to it on a reliable online source that we can check it. Another small point. You say you are attracted to "thirderism": I hope you realize that if you really believe in your argument then you can similarly "disprove" "thirderism" as well. So you cannot consistently be a thirder. Thank goodness your argument is flawed. Oh, well… you can't have it both ways. • Michael says: @APradeep You write: "From the fact that p(heads|Sunday) = 1/2, and p(heads | Wed interview ) = 1/2, SB can rightly conclude that in this period, she never received any direct or probabilistic information about how the coin actually landed when tossed. Even on Monday and Tuesday, she cannot have had any information about this. So when SB is asked about the subjective probability that the coin actually landed heads in the act of tossing, she can therefore truthfully answer 1/2 even on Monday and Tuesday." I disagree. On Monday and Tuesday she in fact has the information that the experiment is in progress. So she calculates p(heads|expt). This is the requested subjective probability. To put it another way, if she "truthfully" answers 1/2 on Monday and Tuesday, then she is ignoring the crucial information she has, that it is Monday or Tuesday. Ignoring this information has real negative consequences (e.g., if she acts on a belief of 1/2, she will incorrectly expect to win on average by accepting a bet where she gets$15 is the coin is heads and pays $10 if the coin is tails). So I'm afraid I have to interpret the above logic of "truthful answering" as being equivalent to "lying to herself about what she actually knows". The SB problem unambiguously asks for p(heads|exptl interview), and it is therefore simply incorrect (or, at the very least, not rational) to give the answer p(heads|exptl interview) = 1/2 (and similarly incorrect to justify it on the grounds that p(heads|Wed) = 1/2). Still, it's her money to lose, I suppose. More generally, I feel that telling people "It's OK if you said 1/2 for the SB problem, because 1/2 is indeed the answer to another problem", is akin to telling children "It's OK you said 1+1=1, because would indeed be valid if 1=0". It might make them feel better, but it won't teach them anything useful about how to calculate in the real world. • Brett says: What you call "a lot of irrelevant stuff" I call "trying to make sure it's clear–to both of us–where exactly we disagree", but OK. First: nothing I've said implies that thirderism is false. Your thinking otherwise demonstrates a lack of understanding of the basic issues in this area. (See, I can be dismissive and insulting too! Doesn't mean I should be.) Here are two distinct questions: 1. What is the objective chance of the coin's landing heads? 2. What is your degree confidence that the coin landed heads? It's very obvious that these can come apart. The objective chance of a fair coin's landing heads 1/2, of course, but my confidence can differ depending on my information state. If, for instance, someone (who isn't a known liar) watches the coin toss and then says to me "The coin landed heads", my confidence that the coin landed heads should be significantly higher than 1/2–despite the fact that the objective chance of the coin's landing heads is 1/2. Similarly, in the Sleeping Beauty case, halfers and thirders can, should, and do agree that the objective chance of the coin's landing heads is 1/2. But the answer to the question how confident she should be that the coin landed heads–and this is the question asked in the Sleeping Beauty problem, unambiguously–is not determined by facts about objective chance alone; it depends also on what information she has. The disagreement between halfers and thirders is about whether the information that she is currently awake on either Monday or Tuesday can, by itself, have any bearing on what her degree of belief should be that the coin landed heads. Second: though I'm going to play along, I'm not sure it's my job to direct you toward the materials you're asking for. Since you're the one stating as fact that every probability theorist who has thought about the Sleeping Beauty problem up to this point has been confused about something trivial, maybe the onus should be on you to at least learn the axioms of probability theory and their immediate consequences. In any event, here's an online source that makes the point briefly: https://www.otexts.org/node/129 Here's another, from a textbook published by McGraw-Hill, with more explanation: http://fitelson.org/bayes/schaums_logic_ch10.pdf Unfortunately, some of the relevant pages here are missing for some reason. The axioms are stated on page 2 of the PDF, but the proof of the claim we're talking about is on one of the missing pages, though you can see that the authors refer back to the theorem–they call it "Problem 10.6"–repeatedly in the course of other proofs. It goes something like this: Suppose A and B entail each other. Then A and ~B are mutually exclusive, and similarly for ~A and B. So, by AX3, Pr(A v ~B) = Pr(A) + Pr(~B), and Pr(~A v B) = Pr(~A) + Pr(B). Furthermore, given the entailments, both ~A v B and ~B v A are tautologies, and so, by AX2, Pr(~A v B) = 1 and Pr(~B v A) = 1. So by simple algebra, we have Pr(A) + Pr(~B) = Pr(~A) + Pr(B). Finally, A and ~A are mutually exclusive, and A v ~A is a tautology (and similarly for B and ~B). So by AX2 and AX3, P(A) + P(~A) = 1, and P(B) + P(~B) = 1. Now all that's needed is some more simple algebra to see that P(A) = P(B). Third: given all that, there's really not much to say about your counterexample, but I'll try to explain briefly why your description of it is confused. You're conflating the proportion of items in a set that meet a certain description with the probability of an event. These are, of course, not the same thing. I have committed no "cardinal error", and "Pr(2) in PickB" is not a phrase that actually makes any sense. "2" is not a proposition. What's the actual proposition here? I'm assuming you mean "Pr(Q) in PickB", which also doesn't really make sense, because Q is a proposition about what's picked from set B, not a proposition about whats picked from PickB. Or perhaps you mean "Pr(2 is picked from PickB)". But on any reasonably interpretation of P, that proposition and P don't entail each other, so there's no counterexample here. I can't really continue this discussion–it's taken up far too much time already–but I encourage you to consult an expert on probability theory. • Pradeep Mutalik says: @Michael, The answer 1/2 is not an answer to a different problem statement, it is an answer to the same problem statement, differently interpreted. Let's consider my linguistic example. I have a friend who is a rock star and makes 10 million dollars a year today. Twenty years ago he made$20,000 a year. I tell you, "My friend spent an entire year's earnings to pay for his first guitar." How much did he pay for the guitar twenty years ago?

You as a listener have two choices: You can imagine yourself back in the time period that the conversation is referring to and conclude $20,000, or you can just mentally stay in the present time and conclude$10,000,000. Both are linguistically valid, and you have to figure out what I meant based on context. In everyday conversation, one of the options makes more sense than the other.
In this case you would probably imagine yourself going back to the time referred to, and conclude 20,000. Now look at the SB problem statement: "What is your subjective probability that the coin landed heads?" Does SB imagine herself in the past looking at the coin being tossed as you did in the rock star case (answer = 1/2), or does she mentally stay in the present time and answer based on that (answer = 1/3)? Both are valid interpretations linguistically. I agree with you that if the SB problem was given to you in a math test with the intent of testing your probability skills then you would assume that it is the second answer that is expected, because otherwise the details of the experiment are irrelevant. But on the other hand maybe your math teacher is one of those smart aleck jokers, who prefers the first answer. "I gave the problem just to see how carefully you all read the question: all the details are just distractions." The point is that both interpretations are valid. Now I don't mean that people think this way consciously. They just make a choice about how they are going to solve the problem and fit in their interpretation based on that. I have found that thirders seem to be people who are pragmatic, look at things in perspective from where they are, and naturally think in terms of bets and frequencies. Halfers seem to be more idealistic, concerned with information obtained about what actually happened in the world, prefer the "view from above" and naturally think in terms of conditional probabilities. Thirders count every wakening as a separate event. Halfers count every experiment as a separate event. I laid out these two approaches in my discussion of this problem back in January. My position now is that most halfers and thirders are considering different valid propositions based on the problem statement, they are both right and there is no mathematical conflict. It's just a conflict of style and interpretation. • Pradeep Mutalik says: @Brett OK, let's agree to disagree and part cordially. Let's forget and forgive all instances of snarkiness on both sides. I think the crux of our disagreement is that the point you are making seems to refer to a single probability space (where the probability measures of the possible outcomes add up to 1), whereas in my proposal there are two probability spaces, each of which has a set of different probability measures that add up to 1. There are two different probability functions. So you cannot do the "simple algebra" that is described. —– If you take my two propositions that you called P1 and P2: P1: The coin landed heads in the act of being tossed. P2: The preserved coin is showing heads now. Now change P2 to "I am currently in a heads timeline" or "The coin in my past landed heads." It seems to me that you can make the same argument against the thirder position. In my explanation, the "coin in the safe" was always just a mental image to get a better handle on what timeline you were in —– You are free to comment or not. If I make any comment it will be brief and I promise, cordial. Best, Pradeep • Louis Wilbur says: The sleeping beauty problem is ambiguous because it does not say what sample space she is using. Probabilities are defined on a per sample space basis. The sample space of the coin toss is {H,T} and the sample space for the questions about the coin state is {MH,MT,UT} where H=heads, T=tails, M=Mondays and U=Tuesdays. The probability of heads for the first sample space is 1/2 and the probability of heads for the second sample space is 1/3, since they are both equiprobable sample spaces. To see equiprobability, just notice that out of every 1000 coin tosses about 500 will be heads, 500 will be tails, and about 1500 questions will be asked about 500 which will occur when it is Monday and heads, another 500 which will occur when it is Monday and tails, and the remaining 500 which will occur when it is Tuesday and tails. She should use the probability for the sample space she assumed and the problem doesn't tell what sample space that is. The problem is bad because it introduces two different sample spaces without clarifying which one is operative. For example, if the problem also stated that for betting purposes on repeated trials of the experiment, with payment made on a per question basis, then she should bet as much money as possible and it would be clear that she should use the sample space for the questions about the coin state to get the probability. But if instead of that we added to the original problem that she give the probability for repeated tosses of the coin then money won or lost during each question is irrelevant and she should use the sample space for the coin toss to get the probability. The sleeping beauty question is ambiguous because it is asking about belief in the frequency of the truth value of occurrences of the PROPOSITOIN "the coin landed heads" not the proposition that the coin's probability of landing heads is 1/2. That is, the question doesn't make clear if it is asking about the probability of the proposition being true during repeated coin tosses or if it is asking about the probability of the proposition being true during repeated questioning in many repetitions of the experiment. These are not the same thing because when the coin is tails she is questioned twice but when the coin is heads she is questioned only once. Now she knows the proposition is true one out of every three times she is asked and she is not going to mistake that for the fact that the coin comes up heads one out of every two times during coin tossing. So, for the proposition "the coin landed heads" the frequency of this proposition being true during repeated questioning in many repetitions of the experiment is different than the frequency of it being true during repeated coin tosses. If the coin toss actually came up heads then the proposition "the coin landed heads" is true but if the coin toss actually came up tails then the proposition "the coin landed heads" is false. How often the proposition is true or false depends on the circumstances. So adding two different prepositional phrases onto the original question highlights the ambiguity of that question: (case 1) What is your belief now for the proposition that "the coin landed heads" in the case of repeated questioning in repetitions of the experiment? (case 2) What is your belief now for the proposition that "the coin landed heads" in the case of repeated tosses of the coin? The conclusion: the sleeping beauty problem is ambiguous because case 1 and case 2 use different sample spaces and if one removes the phrase "questioning in repetitions of the experiment" from case 1 and removes the phrase "tosses of the coin" from case 2 then the ambiguity of the original question is exposed. • Pradeep Mutalik says: @Louis Wilbur, I agree with you completely. Ambiguity is the problem here, both linguistic (two interpretations) and mathematical (two propositions, two styles). But there's something that complicates the problem further. When halfers count experiments, they assign probabilities of {1/2,1/4,1/4} to {MH,MT,UT} respectively, which conflicts with the equiprobability of the statistics. One more thing to add confusion. • Wiley says: @Pradeep, apologies for not responding sooner. Re: your comment of being able to keep all 5 propositions in my earlier comment and your response to eJ and Brett above: Probability space is built on formal systems, but it is also subject to interpretation. There are indeed interpretations in literature that fail to satisfy all of Kolmogorov's axioms. If this were intended in your arguments, I would not be able to say anything intelligent since I know close to nothing about these theories. But if you agree with the premises of Kolmogorov's axioms (hopefully), then your claims are mathematically wrong, e.g., given P iff Q, P ∨ ¬Q is a tautology, so Pr(P) + Pr(¬Q) = 1 from the finite additivity axiom; similarly Q ∨ ¬Q is another tautology thus Pr(Q) + Pr(¬Q) = 1, which results Pr(P) = Pr(Q). However, you could have two separate probability spaces and claim that SB is free to interprete either one as her subjective probability. This goes back to the point that mathematically both models are consistent, but the philosophical controversy remains, which I suspect is what you would like to focus on. Re: betting analysis: Dutch book arguments are traditionally associated with subjective probability. It does give an operational definition of credence, but what has been missing in the comments here is what type of bets are considered to be "fair". In particular, halfers could argue that SB knows that she would be betting on the exact same T twice, while once on H, thus it's the payoff rather than her credence that leads to 1:2 betting odds. There are numerous literature on this with Dutch books against either halfers or thirders. Besides Bradley and Leitgeb (2006) in my previous comment, I personally find Hitchcock (2004) and Lewis (2010) are worth reading. This could be a good point to emphasize the importance of Kolmogorov's axioms. One is free to ignore all the "fairness" constraints and use the literal betting odds to define subjective probabilities. It simplifies the definition, but there would be no guarantee of mathematical consistency. On the contrary, if your subjective probabilities obey the axioms, there exists no Dutch book against you (Kemeny 1955). Few tangential points: (1) One "natural" way to incorporate the temporal element of the experiment is to model SB's situations with a Markov chain. Suppose the experiment is being repeatedly carried out every week. We have the following three discrete states that are relevant for SB: (H1) H and Mo, (T1) T and Mo, (T2) T and Tu, with transitions: (1) H1 -> H1 or T1, (2) T1-> T2, (3) T2 -> H1 or T1. The transition matrix is [[1/2, 1/2, 0], [0, 0, 1], [1/2, 1/2, 0]], which results a stationary distribution of [1/3, 1/3, 1/3]. However in order to account for different amount of times that SB stays in these states (it takes twice as long e.g., from H1 to T1 as from T1 to T2), we need to use a continuous Markov chain. The transition rate matix is [[-1/4, 1/4, 0], [0, -1, 1], [1/2, 1/2, -1]] and the stationary distribution becomes [1/2, 1/4, 1/4]. (2) There is a recent longreads article on wired.com featuring a woman who has no so-called "episodic memories". Maybe she is an ideal subject for the SB experiment: http://www.wired.com/2016/04/susie-mckinnon-autobiographical-memory-sdam/ Reference: Hitchcock, C., 2004, "Beauty and the bets" Kemeny, J., 1955, “Fair bets and inductive probabilities” Lewis, P., 2010, "Credence and self-location" • Pradeep Mutalik says: @Wiley, Thanks Wiley. A scholarly analysis as usual. I think you have arrived at the same conclusion as I did regarding Brett and my disagreement. My proposal clearly has two distinct interpretations with two different credences and two probability spaces. What did you think of Groisman's paper? • eJ says: Just to keep my hand in, so to speak, recapping a couple of ways I diverge from Wiley's analysis (detail is in earlier posts): 1. Probability spaces. I don't think Credence acts like a probability. In particular, I can ascribe credences {1/2,1/4,1/4} to the wake-up situations, yet also 1/2 to "Heads given Monday" — in disagreement with 2/3 from Bayes theorem. 2. Dualism: "you could have two separate probability spaces and claim that SB is free to interpret either one as her subjective probability". I don't think a rational agent is allowed to flip on a whim like that. As puzzle readers, we tend to, though (and this is the deep and interesting root of Halfer vs Thirder tribalism). • Marcel says: SB knows how the experiment is done, because she is told. So she knows the chance the coin will land heads is 1/2. When she is awakened she doesn’t get information on top of that, so her best guess is then 1/2. This would also be the case when she wouldn’t be awakened two time in case of tail, but a billion times. But when SB would have been told: by some process the outcome of the experiment will be heads or tail, the situation is different. The only information SB has about the outcome of the experiment when she is awakened, is the fact that she is awakened and her best guess would be 1/3. • Pradeep Mutalik says: @eJ, In my proposal, Beauty is not flipping on a whim. She simultaneously has two rational credences about two different questions that the problem can be teased into. I think that a lot of these divergences of probability vs credences and credence vs Durch Book arguments were 'invented' specially to deal with the SB problem. My proposal (actually Groisman's) shows that these exceptions are unnecessary: credences, probability and Dutch book arguments can be realigned here as they are elsewhere, without a conflict. We can return to being purists. • Peter says: I'm sorry but it's 50:50. If SB was to be woken on Tuesday she would think a) H-M it's Monday maybe or b) T-M it's Monday maybe or c) T-M its now Tuesday maybe. b & c are the same as there was no coin flip on Tuesday, where b & c make reference to only the coin flip on Monday. • Peter says: How else would she answer, she's going to physically say 50:50 heads or tails on Monday and I don't know what day it is. She wasn't asked to gues the day, if she was asked to guess the day as-well then and only then the experiment would change to 1/3 on Tuesday. So the answer is 1/2. • eJ says: Pradeep, I'm sticking to my argument of April 14, 2016 at 6:34 pm … I am quite sure that it's irrational for SB to consider the logically-equivalent P1 (coin landed Heads) and P2 (coin currently showing Heads) to be separate propositions. P1 and P2 come without any attached probability space: they are simply statements to be evaluated in the context of a probability space determined by other means. In Beauty's case, the probability space is determined upon learning the rules of the experiment; in the case of puzzle readers who are a bit Halfer/Thirder flip-floppy, the probability space is intuitively influenced by the manner in which the proposition is phrased. • eJ says: PS: when is the panel of experts going to weigh in? We have surely exhausted all productive amateur commentary … • Pradeep Mutalik says: @eJ, We haven't received any replies yet. We will continue to try, but this proposal probably needs to be submitted to a professional journal to get the attention of the community. • Wiley says: @eJ, As mentioned in my first comment of this thread, I prefer to be a double-halfer, but I feel like it would digress the topic further to explain the role of indexicality when applying Bayes' theorem. @Pradeep, Unfortunately I am in no position to critique Groisman (2008). I could however add few words about my personal view of the problem, which should by no means be considered as criticism of anyone's work. The SB problem to me is mathematically uninteresting, mainly due to the fact that we have two simple consistent probability space with no further restriction. It's a completely different situation to philosophers since there are more constraints, specifically SB should be rational. To me as an outsider, if it were to be the case that some of these cherished principles of rationality are sufficient to constrain her credence as how the problem is stated, wouldn't it be more interesting than devising experiments to claim that either probability space is viable, a fact that is already known? More importantly, there are pratical cases that one simply cannot run these "well-defined" experiments repeatedly. Anthropic reasoning is one such example as our observation of the universe is limited by the event-horizon. (In the field of cosmology, there is controversy whether anthropic principle should be considered to be real science or not, but I always find Steven Weinberg's prediction of cosmological constant fascinating) • Percy Forest says: At the start Sleeping Beauty believes three things – 1) the coin is fair, 2) she will be awakened once after a toss of heads and twice after a toss of tails and be offered a wager each time, all the wagers being the same, 3) when she is awakened she will not know the day. It is neccessary to know the objective value of P(H) in order to successfully decide whether to accept the wagers. After the experiment begins, what might she learn? And what might she update? Referring to the original problem – 1) she might learn that it is Tuesday, which is logically equivalent to the coin having fallen tails, 2) she might learn that the coin fell heads, which is logically equivalent to it being Monday, 3) she might learn that it is Monday, which tells her nothing, being consistent with what she believes. In no case does she need to alter her belief that the coin was fair. Fair coins can fall heads and tails. If, as in variants of the problem, the number of awakenings or wagers is changed her decision to accept the wager may change. Specifying the days is a redundant way of giving the number of wagers after heads (tails). A functionally equivalent 'compressed' version of the problem is: she will be interviewed once and offered a wager after a fair coin is tossed; she will be paid double after a toss of tails, but not after a toss of heads. There are no bells and whistles here – no 'epistemic possibities', no induced amnesia. She will have no reason to change her belief that the coin is fair and no reason to believe that it landed one way or another. Assigning probabilities to the 'epistemic possibilities' is a gigantic red herring which forces one to model the problem as if it were a shell game. Thirders and halfers differ only in how they assign probabilities to the pea being under a given shell. Betting considerations are completely irrelevant with respect to her belief about the coin, but they are a wonderful distraction – which, along with the 'epistemic possibilities', reinforces my belief that this all started as a practical joke. It really is that simple! • Louis Wilbur says: If the experiment is repeated one thousand times then, on average, she will be awakened and asked about her belief fifteen hundred times during the experiments. On average, five hundred of those fifteen hundred times she is asked (i.e. one third of the times) the coin will have landed heads and so the proposition "the coin landed heads" will be true. On average, one thousand of those fifteen hundred times she is asked (i.e. two thirds of the times) the coin will have landed tails and so the proposition "the coin landed heads" will be false. So sleeping beauty knows several facts. She knows that in repeated trials of the experiment, on average, one out of every three times she is awakened the proposition "the coin landed heads" will be true. She knows she was just awakened. So it is reasonable for her to give one third as an answer. She also knows that in repeated trials of the experiment, on average, one out of every two times that the experiment is run the proposition "the coin landed heads" will be true. She knows the experiment is being run. So it is reasonable for her to give one half as an answer. Her answer to the question should be: My belief is that, on average, the proposition is true one out of every two times the experiment is run and, on average, one out of every three times I am awakened and asked my belief about it. • Bob Lince says: Consider: A fair coin is fairly tossed a number of times. Every time a head shows, a photo is taken showing the "heads" clearly. Every time a tails shows, two photos are taken, each showing the "tails" clearly. All the photos are turned upside down and efficiently randomized. You are brought into the room and told everything that has happened. Two questions: Q1) What is your degree of believe that a random toss of the coin showed heads? A1) Given the conditions, about 50%. Q2) What is your degree of believe that a random photo, when turned over, will show a "heads"? A2) Given the conditions, about 33%. Halfers and Thirders agree on the answers to both questions. They disagree on which question is being asked. But that disagreement is caused by a lack of precision in language, not in mathematics or logic. For the Sleeping Beauty problem, is the question: SBQ1) What is your degree of belief that a random toss of the coin showed heads, or, SBQ2) What is your degree of belief that on a random morning, being woken up, the cause of your being woken up was that the coin showed heads? • Trevor Redding says: Put me down as a "halfer", because you have to determine the differing probabilities of SB being awakened on a Monday or Tuesday. On the Monday, the probability of SB being awakened due to a heads is 2/4; being awakened due to a tail is 1/4. On Tuesday, the odds of being awakened due to a tail is 1/4. Therefore, the odds on it being a Monday is 75% as compared to 25% for a Tuesday. To generalise, when you compute the probability of SB being awakened on a specific day the paradox disappears, and SB should always reply that her belief in the coin being heads is 50%. • Bob Lince says: If the experimenters wake Sleeping Beauty on Monday due to Heads showing they send you an email saying "Heads". If the experimenters wake Sleeping Beauty on Monday due to Tails showing they send you an email saying "Tails". If the experimenters wake Sleeping Beauty on Tuesday they send you another email saying "Tails". Q1) What are your chances of getting an email that reads "Heads"? A1) 50%. Q2) Of the email you get, what are the chances of an email that reads "Heads"? A2) 33%. Is not "Which is the question?" the question? • Steve says: I started off as a halfer, but end up a thirder. I think it is less a wording issue, more a question of whether is using all avialable information available to infer prior events. If SB looked at the coin toss as a stand alone event, SB will be a halfer for sure. However, since SB knows the imbalance rules of the game, SB should take that into account to infer the optimal result. In which case, thirder is the only answer If we change the game a bit such that, after a single fair coin toss, heads only 1 person is brought in and asked, "do you think the toss ended up in heads or tails?" Tails, a new person is brought in each day and asked the same question for eternity. If you were brought in one day, which answer would you provide? In this case, given the imbalance nature of the trials, tails is more likely to be correct due to the imbalanced nature of the game. (Since with head the game terminates as soon as the first person is questioned.) • Trevor Redding says: @ Bob Lince In your email analogy, the question is "what is SB's belief that of the 3 emails presented, she will pick an email reading heads". You say 33%, because there is an equal chance of picking each email. I say 50%, because I believe that the pick is biased, and SB has twice the chance of picking a "heads" email. The equivalent in the "real" story is that you should assign probabilities in a two step procedure:- 1.What day is it likely to be? 2. Given the day, what likelihood of heads or tails? There is a 50% likelihood of heads, so 50% likelihood of Monday. There is 50% likelihood of tails, so 25% likelihood of Monday and 25% likelihood of Tuesday. So the chances of it being Monday is 75%; that of Tuesday is 25%. For the "thirders" argument to work, the chances need to change so that the chances of it being Monday is 66% and that of Tuesday is 33%. • Heiko242 says: If the SBP is actually due to an uncertainty in lingustics, as speculated above, it might be sensible to formulate problem within a more formal way. E. g. within the framework of modal logic, or formal semantic languages (Montague grammar, Kripke semantics?). Frmo this formal semantic approach the logic contradiction maybe derived analytically. Can analytical philosophy help? • S Galloway-Walker says: I am aware that I have come to the discussion very late and it may be that someone has made a similar point previously. I don't have a week to spare to read thru the discussion. To my mind it's pretty straight forward and complicating the "puzzle" by introducing poisoned beans etc is just adding to the confusion. Instead, consider an alternate but hopefully simpler scenario. SB sits at a table where a coin is being tossed and by some means (drugs and awakening, shutters or whatever) is only permitted the witness one heads outcome in every 10 tails outcomes. If asked what is the likelihood of a heads outcome she will of course say it seems pretty likely. That is her subjective view based on being presented with a biased sampling of what actually takes place. All that demonstrates is that subjective views of probability are subjective. I don't see how the puzzle as presented is substantially different from this scenario. • Trevor Redding says: This is my final post for the halfers; trying to make the argument clearer! You should assign probabilities in a two step procedure:- 1.What day is it likely to be? 2. Given the day, what likelihood of heads or tails? There is a 50% likelihood of heads, so 50% likelihood of Monday. There is 50% likelihood of tails, so 25% likelihood of Monday and 25% likelihood of Tuesday. So the chances of it being Monday is 75%; and if awakened on the Monday the likelihood of it being heads is twice that of tails (as 50% is twice that of 25%). So SB assesses the probability of heads as 2/3rds of 75% =50% (from Monday only) and that of tails as 1/3rd of 75% (from Monday) plus 25% (from Tuesday) = 50%. Thus SB has an equal belief in heads or tails. • Steve Cheung says: Thinking about this more, I actually think that describing this an a linguistic or interpretation issue. It's a question of how to treat the problem is more rational. Even reading the question on the face value, there are two ways to deal with the issue. If we take the approach I mentioned above—which I learned is named Extreme Sleeping Beauty by Nick Bostrum since I typed my comment— where the we have a new subject being brought in everyday for questioning for n days if it is tails, where n > 1 and n is an arbitrarily large number. Knowing the rules, the subject is faced with a choice: 1. Ascertaining the probability based on the probably of the event itself, i.e. Halfter approach 2. Ascertaining the probability based on the probably of the experiment, i.e. Thirder approach, or in the extreme case, Zeroer approach. The crux of the matter, is not really an interpretation of the question, but a choice of how to approach the problem in an ambiguous(from the perspective of the subject) situation. Using the extreme SB approach, i.e. making n approach infinity, makes the problem interesting because it amplifies the difference. Yet a rational choice, should work for both extreme SB and the traditional SB problem. So which approach is rational for the subject? If the subject choose approach 1, i.e. the Halfer approach, the subject is ignoring the property of the experiment itself, so it is less than ideal. If the subject chooses approach 2, saying that a fair coin toss has nearly 0 chance of leading on heads, just because more subjects are interviewed also seems less than idea, especially since the subject doesn't have any information on how many subjects were interviewed. This becomes even more apparent if we tweeted the parameters a bit further, and say that m subjects are interviewed if the fair coin toss landed on heads, and n subjects are interviewed it landed on tails. Where n > m by an large amount. In the heart of the problem is not an linguistic issue, it is how to choose an approach as the subject in a world where the information is imperfect. Which approach is rational? • Trevor Redding says: Here is a simple proof of why the SB problem can be unambiguously solved; and that the problem is that the wrong assumption is made. Assume that SB is only awakened if the coin is tails. SB will then be awakened 50% of the time on Monday and 50% on Tuesday; whilst she will be left sleeping 50% of the time. The wrong assumption is that the chances that SB will be awakened is chance of being awakened on Monday plus chance of being awakened on Tuesday = 50% +50% = 100%. The correct chance that SB will be awakened is chance that SB is in Monday when awakened plus chance that SB is in Tuesday when awakened =25% +25% = 50%. Similarly, if SB is only awakened on Monday when the coin is heads, the chance that SB is in Monday when awakened is 50%. Combining the assumptions, the chance that SB is in Monday when awakened is 75%, and the chance that SB is in Tuesday when awakened is 25%. From above, SB can then compute the chance of a head as 2/3rds of 75% =50%; and the chance of tails as 1/3rd of 75% plus 25% = 50%. • Dominic Tremblay says: Making it short: P = (g)n-1 [Except the first occurence] -> In this case 1st group becomes irrelevant in an entangled state with a possibility or 2 or 3 items, so you have not to consider the first… so only 2 both sides = 1/2 = 1/2 So 1/2. • Bob Lince says: Perhaps apropos this article, the Colombian aphorist Nicolás Gómez Dávila wrote: "A certain intellectual courtesy makes us prefer the ambiguous word. The unequivocal term subjects the universe to its arbitrary rigidity." • Trevor Redding says: The thirders make two errors. The first error is to fail to ensure that their probabilities total to unity. The only way to get 1/3rd and 2/3rd probabilities is as follows. If tails, SB will be awakened 50% of the time on Monday and 50% of the time on Tuesday, so chances of SB being awakened by tails is 50% + 50% = 100%. If heads, SB will be awakened 50% of the time on Monday. So total chance of SB being awakened is 150%; and chance of heads = 50/150 =1/3rd and chance of tails is 100/150 = 2/3rds. Obviously, this logic is flawed as the chances of Sb being awakened must sum to 100%. The second error is exposed if we assume that SB is only awakened by tails; she is left sleeping if the toss is heads. The chance of tails and SB being awakened is then 50%. If tails, you can wake up SB once, twice, or a hundred times, but it has no affect on the chances of heads, and the probability that SB will be left sleeping remains at 50%. • eJ says: Slightly off-topic, here's a simple variant of SB in which "the coin landed heads" and "the coin is currently showing heads" are different propositions for which the credences-for-Heads could reasonably be claimed to be 1/2 and 1/3 respectively. The variant lasts two weeks. Week 1 is per normal SB. At the following weekend, Beauty sleeps through (so she never knows what week it is) and the coin is turned over to show the other side. Interviews in week 2 are then as dictated by the turned-over coin. It occurs to me that the "coin currently showing Heads" question might split Halfers: some will stick rigidly to 1/2, others will say 1/3. I'm drawn to this because of the Limbo example in Masahiro Yamada's "Laying Sleeping Beauty to Rest" paper in which, to my surprise (as I agreed with much of what he'd written prior to that), the author in effect plumps for 1/2. • Steve Cheung says: @Trevor The thirder is not making a mistake. Halfer describe the problem in the most basic way, which is the probability of the coin toss itself. Since the coin toss is 50/50, no matter how you break in down it will be 50/50. However, doing that is like playing blackjack without looking at what cards have been dealt in previous hands, and how many hands are dealt before the cards are shuffled. By looking beyond the probability of the deck, and looking at how rules and process affects the game one can potentially gain an edge, like card counting. Thirder is looking at the probability of the game to infer what's more likely to be the case. Since there are two occasions that the subject is questioned if the original toss was tails, the probability of the being questioned is 1/3=> heads, 2/3=> tails, which adds up to 1. It is not looking at the original coin toss' probability, but looking at the likelihood of someone being asked a particular question, then inferring the result of the original event. The analogy with card counting breaks down, however, cause SB doesn't have prior knowledge that card counters do, and tails has an edge based on expected values. However, expected value doesn't really work, because SB loses her memory, so in effect, she only ever gets one chance. That's where the dilemma comes, and the heart of the argument I believe. • Trevor Redding says: @Steve SB buys your argument! The procedure is to be run 100 times so, to make it worth her while, SB demands1 each time that she guesses the coin toss correctly. SB figures that, on your reckoning, if she always guesses tails that she will be correct 2/3rds of the time and will win, on average, $67. I tell her it will only be$50, on average. Which one of us is correct?

• Steve says:

@Trevor Neither. The rules are unclear in that case. Is SB betting each time she asked the question or is she betting on the overall experiment? I.e. if the coin toss was tails, and she's asked the question twice, she only gets she same reward as if the coin laded heads and she was only asked the question once? If each time she's asked the question, she gets rewarded, thirder is right. But if SB only gets the same reward no matter how many times she's asked the question, you would be right.

Interestingly, the edge in each of these cases lies on different sides. In the case where the bet is on each time the question is asked, the edge on for tails. In the case where, there is only a single reward, the edge is on heads. (With tails, SB has additional risk for answering the question differently each time she's asked.) But I digress.

The SB problem has a number of issues inherently. First, the question asked to SB is the exact question we are pondering, which is the right way to ascertain the "degree of certainty"? The actual question is not something can be bet on since we don't have the right answer. Knowing the actual result of the coin toss does not help answer the question.

Second, we inherently want to convert it into a bet to illustrate either thirder or halfer will get a higher payout. When we do so, we are not clear what is considered a trial. We can define a trial as either time SB is asked the question, or one iteration of the complete experiment. Since that is not clear, each side, halfer and thirder take their definition as the correct one, and no one will ever agree.

Now thirders would say, if we treat the experiment as a whole, it cause all kinds of problems because of the potential for how inconsistent answers is treated. So treating each time SB is questioned as a trial is more correct.

Based on that position, we get into another problem and unclear definition. Is the payout for getting heads correct once equal to getting tails correct once? Or is the payout for getting head correct once equal to getting tails correct twice?

Each of us, when we trying to calculate the payout, and which position make more sense, are implicitly filling in the blanks of the problem in our reasonings. For me, I like the thirder's position better, but it doesn't really mean it's more right. I like it because it uses to the rules to infer a past result, just my personal preference. Inherently, if my definition of trials and payouts are different that yours we will arrive at different conclusions.

• Stephen Moratti says:

I cant understand how anyone could think it was 50:50. If She was awoken a trillion times when it was tails, but only once when heads, then when she was awoken sometime she would be silly to think it might be heads.

• Percy Forest says:

@Stephen Moratti – Suppose the coin were heavily biased, say ten trillion times more likely to fall heads. Then even if she were awakened a trillion times after tails and only once after heads she should bet heads. To know this she must know and believe the actual probability which governs the toss, just as she must believe the coin is fair in your (actually Bostrom's) example. The proper betting strategy depends on the actual probability which governs the toss, and her belief must track the actual probability. Inferring her credence from the betting strategy is also possible with repeated experiments, but it is a mistake to identify it with the ratios of outcomes from repeated trials. In short, her credence and her betting strategy are quite different things.

Final Words

In this column, I presented the point of view that there are two senses in which the Sleeping Beauty problem question can be understood, and that the two senses have two different answers.

The two questions are:

1. What is the subjective probability of the coin having landed heads when it was tossed?
2. What is the subjective probability of the coin having landed heads in SB’s past when she is awakened? (Note that the "coin in the safe" is just a proxy for how the coin in SB’s past landed).

The difference between the two senses is subtle, and it is easy to think that they are the same. Some commenters here did understand the difference between the two senses, and a few expressed the difference in their own way.

The version of the questions that was found most useful by other commenters is that expressed by Pace P. Nielsen.

#1: "What is your degree of certainty that, at the flipping, a heads was flipped?" Here, it is irrelevant that the scientists are making twice as many measurements in the tail's case. Those measurements have nothing to do with the actual flipping. The answer is 1/2.

#2: "What is your degree of certainty that this is a wakening with a heads flipped?" Note, this question is asking about more than merely the state of the coin, but also the state of affairs in the experiment. The answer here is 1/3.

The sticking point for some objectors, including eJ and Brett here, is that they think that my two questions are the same: that they refer to the same event, or that their probabilities are the same because they imply each other.

It is easy to show that they do not refer to the same event. The coin can be tossed without SB being involved at all, maybe before she is even chosen as a subject. The sample space for the coin being tossed is {H, T} with the probability of heads being ½. This question can be put to anybody, you or me or an innocent bystander, and unless the people being asked the question have any knowledge of how the coin actually fell in the experiment being referred to, they will have to say that their subjective probability for heads is ½. The result of the coin toss defines whether the experiment will be filed as a “heads experiment” or a “tails experiment.”

The second question can be put to SB and SB only, whenever she is awakened under the experimental protocol, and this can happen more than once. Now, it is true that the result of the coin toss implies whether heads or tails took place in SB’s past — eJ and Brett are right about that. But that does not mean that the probabilities asked for in the two questions are the same. That’s because the question is being asked on potentially multiple occasions, so the sample space for this new question is different. It is now {HM, TM, TT} with a probability distribution of {1/3, 1/3, 1/3}, so the probability of heads in SB’s past on awakening is 1/3.

Note that we can manipulate the probability of SB waking up to a heads coin past to any value between 0 and 1 by simply designing the experiment to wake her a different number of times on heads and tails, regardless of the result of the coin toss. Thus if SB were to be woken up a million times when the coin landed tails and only once when it landed heads, her probability of waking to a heads past would be 1 in one million and one, even though the chance of the coin landing heads when tossed is still 1/2. This can only happen because the sample spaces and probability distributions for the two questions are quite different.

How can the probabilities for the two senses of the question in the SB problem be different in spite of one event implying the other? It’s because there are two sets of implications:
H implies HM.
T implies TM and TT.

So the original probability distribution of (1/2,1/2) neatly maps to the different probability distribution of (1/3,1/3,1/3) for this second, different question.

Thirders are answering the question about what the coin showed in SB’s past when she is awakened. Note that the answer to this question is completely controlled by the ratio of how often SB is woken up relative to the heads or tails result of the original coin. Emphatically, thirders are not answering the question of what happened when the coin was flipped. Surely, no thirder believes that if the experiment were done 300 times, there would only be 100 “heads experiments.” The coin is fair and so there will be as many heads experiments as there will be tails experiments — about 150.

Halfers are indeed trying to answer the question of how the coin actually fell and are influenced by their strong intuition that the answer should be ½. However they are, as a consequence, completely messing up the probability distribution regarding what happens when SB is questioned on awakening about how the coin fell in her past —which as we saw, can be manipulated quite independently from the coin toss.

Halfers do not have to try and find a compatible probability distribution for the three possible awakenings. They can simply ignore them, as they have no relevance to the first question at all. Halfers are making SB answer the question from a universal point of view —as if she were being asked a fact about the world, not a fact about her personal past when she is awakened. It is true that SB gains no knowledge about how the coin fell in the world outside her experimental situation, so her answer has to be 1/2. Her personal past and the number of times she is awakened are irrelevant to this answer.

Here’s a similar distinction between personal and universal points of view giving different answers to the same question. Suppose there were some astronauts circling the moon, and a person on earth were asked “What proportion of the moon’s surface area is currently visible?” The answer from the personal point of view of the person being asked would be 50% — since the dark side of the moon cannot be seen from earth. But the same person could also choose to answer from the universal point of view and say 100%, since the dark side is visible to someone —the astronauts. The personal point of view gives the answer from one’s specific vantage point in space and time. The universal point of view gives the answer from “God’s eye view” — the point of view of all possible observers. One’s personal point of view is irrelevant to this answer. This is exactly the case with the universal sense of SB’s question, whose answer is 1/2.

(The example I give in my column examines the two senses of the question from where SB positions herself in time alone. Perhaps this new example of the personal vs. universal distinction that I give here separates the two senses more clearly.)

When we disentangle the two senses of the problem question in this way it becomes clear that we don’t need to use subjective probabilities (credences) at all —we can use the objective probabilities of the two events which, as we saw above, belong to different sample spaces, with different probability distributions. Of course, as always, subjective probabilities will follow objective probabilities since they are clear-cut.

So I disagree with eJ and Percy Forest that credence is the cause of the halfer/thirder disagreement, or that credence and betting strategy are quite different things. They certainly are not for millions of problems outside the Sleeping Beauty problem, so why should an exception be made here? In my view the problem is that when halfers talk of credence, they only talk about SB’s credence for the coin landing heads at the coin toss (Question 1), which they rightly conclude is ½. But there is also another credence: her credence for waking up in a situation where the coin in her past landed heads (Question 2). Her credence for this is 1/3.

I had written this article requesting an “informal peer review” for my Dualist position, and you have given it to me. I am grateful to everyone here who supported my point of view especially Pace P. Nielsen, Jacob, Alan White and Louis Wilbur. I am also grateful to objectors such as eJ and Brett, whose objections helped me crystallize my ideas. The highly original comments of two commenters here, Hiter Peter and rjt forced me to change or expand my original views of certain aspects of this problem. I agree with Hiter Peter that people approach this problem based on their intuitions and mathematical predilections. The linguistic aspects are secondary, and used only to support whichever approach the solver finds more natural. The “time reversal” idea of rjt showed me that the same linguistic ambiguity that I described for the past tense exists for the future tense as well.

I’d like to award a Quanta T-shirt for April to Pace P. Nielsen for his very helpful restatement of my two questions.

Now, at the end of this exercise, I believe more strongly than ever that the Dualist approach to the Sleeping Beauty problem is the correct one. I do not believe that any of the objections have identified any fatal flaw in it. I have set myself the task of writing a serious paper on this topic and submitting it to a professional journal. This may take a while, thanks to my many other commitments, but I will let readers know of the outcome if and when it happens.

Thank you, all.

• eJ says:

We've been having this debate offline, too. It's not my place to quote others, so I'll simply copy my own final words on the subject into this post.

– – – – – – – – – – – – –

You say that your contribution is a linguistic analysis of the problem statement, but you have been very, very, loose with language, arguing mainly by analogy. What follows is an attempt to highlight some particularly problematic omissions.

The question as originally phrased asks for Beauty's "credence now for the proposition that our coin landed Heads". Your linguistic argument is moot if you fail to deal with the language as originally presented. So, you must make clear, does Beauty consider (quoting Dreier):

* the proposition that our coin landed Heads

to be equivalent to exactly one, or potentially either, of (quoting you):

* the proposition that the coin landed heads when it was tossed
* the proposition that the coin landed heads in SB’s past when she is awakened

— I think you're going with "potentially either".

I'd like you to tighten-up the wording on that second proposition, please: it's a bit muddled at the moment. Make it so Beauty can be asked. "What is your credence for the following proposition. X." — where X is a sentence that makes grammatical sense in its own right. (Let's be precise with the language we're arguing about.)

Note "proposition", not "probability": please stick with the original language. Now we must ask how, if at all, the language used to specify such a proposition can define the probability space within which that proposition is evaluated. In particular:

* You think that the probability space is specified unambiguously by the way in which the proposition is phrased.
* Halfers and Thirders each think that the probability space is well-defined, but that it does not depend upon the manner in which the proposition is phrased.
* I think that the probability space is not defined at all: there is no "deeper truth" and no clarity to be had from the language that was used.

I think at this point we're in the realm of linguistic conventions. I don't understand enough about linguistics to argue with any authority (at all!), but it strikes me that reasonable rules for linking "credence" to a proposition include (A) demanding that the probability space be specified e.g. per Groisman's "under the setup of …" clauses; (B) inferring the probability space from the subject of the proposition, in this case the coin (original phrasing) or, more interestingly in Nielsen's wording, either the coin or the wake-up.

Note that your two propositions refer to the coin, without an explicit statement of the probability space, so in case (A) the credence is undefined and in case (B) the credence is one half. Your argument, loose as it is, fails until you can tighten-up the language and address, head on, the link between the proposition as originally phrased and the probability space within which "credence" is evaluated.

And, please, no more analogies, astronomical or otherwise.

PS: I have just discovered that "Dualism/Dualist" is a term used in the related branch of philosophy called Philosophy Of The Mind. That's an unhelpful coincidence. Feel free to pick a less ambiguous word. How about "plain wrong"?

@eJ

(I've already sent this to you by email — but I'm adding a bit here)

I admit that I need tighten up the questions, but Pace and Groisman already do have tighter versions, and I'm willing to go with either of theirs. I guess you don't think their versions are convincing either.
Anyway, I will take up your challenge and run my final versions by you.

How about you? You are a halfer. How would you tightly define what question you are answering?
Also, what is your answer to the Extreme Sleeping Beauty variation (where Beauty is awakened a million times on tails)?
Is her credence for heads still 1/2 or is it one in one million and 1? If it is 1/2, then you are answering the "prior" question — Question 1 above — even if you say you're not.

> And, please, no more analogies, astronomical or otherwise.
How else can we analyze language except by seeing how the same kind of syntax or type of question works in other situations?

• Monte Hall says:

This is my eponymous Monte Hall problem and the answer is 1/3. All objections to this can be reduced to already debunked objections to that problem, which has no controversy among actual mathematicians. Also, royalties owed for stealing my problem can be donated to charity with no hard feelings on my part. Toodles.

• Cauchy says:

If SB goes by a naive Bayesian approach to calculate her credence of the coin landing heads, by evaluating the new information she has gained since being put under on Sunday for the start of her part of the experiment, then she has to answer 1/2. After all, she's gained no new information since waking up: she already knew before going to sleep that she would wake up, so "learning" this fact can't have her update her probabilities.

Similarly, if SB goes by a naive frequentist approach to calculate her credence, by calculating the fraction of times, on average, that that when she's asked the question the coin will actually be heads, then she has to answer 1/3.

I think these are both wrong. Specifically, I believe that the Bayesian approach should not produce an answer of 1/2, and that the frequentist approach should not produce an answer of 1/3. Rather, both approaches should not be able to evaluate a credence. And I claim that the reason why for both cases is because of the amnesia.

To see why the amnesia is the problem for the Bayesian approach, consider the following two variants to the SB experiment:

Variant 1 (V1): The experiment will be carried out as normal, except that SB will not be administered amnesiacs after each day, and of course SB is told this change. Then, if she awakens on Tuesday, she will know she has already awakened once, correctly deduce that it's Tuesday, and be able to conclude that the coin must have been tails. In this variant, I believe it's fairly plain to see that Bayesian SB will answer "1/2" on either of the Monday awakenings, and "0" on the Tuesday awakenings.

Variant 2 (V2): The experiment is carried out almost exactly like the original SB experiment, except that SB is told that V1 is occurring. That is to say, SB is misled about the nature of the experiment that's about to be carried out. The details of the experiment are changed subtly if need be, so that amnesiacs are administered without SB knowing about it. She will go to sleep on Monday thinking that if she wakes up on Tuesday, she'll remember the events of the day and be able to conclude that coin came up tails, but if she wakes up on Tuesday, she'll have no recollection of Monday's events and think it's Monday again. Because she's been duped, the Bayesian conclusion is that she should have a credence of 1/2 on any questioning, even the ones on Tuesday where she's been given an amnesiac.

So now we return to the actual experiment. Upon thinking about V2, Bayesian SB says to herself "Good heavens! When I wake up during the experiment, I have to take into account the possibility that I might have had some of my memories removed, and that my ability to use past information to update my credences might have been compromised!" It's this realization, not the addition of some new information, that should make her credence in the coin being heads change from 1/2. When she's answering questions during the interview, her knowledge of previous events is potentially compromised, and not taking that into account is tantamount to her willingly deciding to partake in V2 rather than the experiment she's actually in.

I don't see how Bayesian SB gets a credence in this scenario. Before the experiment starts, she can say that the probability that she wakes up in the Monday/Heads scenario is 1/2. She might also like to say that Monday/Tails and Tuesday/Tails each have probability 1/4, but these are not indistinguishable scenarios from her perspective, since in one of them her memory's fine and in one she's lost a day of data. And once she's inside the experiment, the two days look the same, but it's possible that her memory's been tampered with! She can't decide here either that the two scenarios are equally likely. Since she can't reasonably assign probabilities to these scenarios, she can't know the chance her memory's been tampered with, so she can't correctly update her credence that the coin flip was originally heads. Of course, if you decide to say that since SB can't assign probabilities to the chance of memory loss, she should ignore it, I guess you still arrive at 1/2, but this seems like a cop-out to me.

The argument against the frequentist conclusion is simpler but subtler. Before the experiment begins, SB knows that if she were to make a 2:1 bet on the coin being heads every time her credence were asked of her (that is, +$2 if the coin is heads, and -$1 if the coin is tails) then she'd come out even on average. However, once the experiment is in progress, things get a lot shakier. The claim is that SB will offer this bet in a given scenario (Monday/Heads, Monday/Tails, and Tuesday/Tails) believing it's even money, because she believes that in the other scenarios she would also offer this bet (and hence actually make even money, on average). But because of the amnesia, these "other scenario" versions of her are neither past versions of her nor future versions of her. She's counting on them making the same wager as her, but the only reason she thinks they'd do that is because they're her in exactly her situation and so would come to the same wager as her. This is a superrational argument, and I reject that superrationality is rationality. (Of course, if you believe that superrationality is a rational conclusion, then you should accept that frequentists arrive at a credence of 1/3.) Certainly, 1/3 is the only possible credence that satisfies the "operational subjective probability" property that's been talked about above, but that doesn't mean that the credence must be 1/3. A stronger argument needs to be supplied to deduce that a credence can actually be assigned (i.e., that SB's not just saying "well, let me try this and see if it works" in the heat of the moment, and finding out after the fact that it did in fact work), and such an argument fails for the reasons I described.

So 1/2 and 1/3 are both not the correct credence, because the credence can't be calculated.

I guess I'd say that the AIP vs. PIP dichotomy seems like a red herring to me. Since the AIP and PIP describe the same coin state in every situation (the coin result back then was heads if and only if it's a heads in the safe now), they must have the same credence if they have any credence at all. The claim of differing credences falls to a "no Dutch book" argument: depending on the frequency at which SB is allowed to bet on the two events, she can make a Dutch book as long as she's allowed to bet on one whenever she's allowed to bet on the other. The differing credences require that SB must bet on the AIP once per experiment, and on the PIP once per awakening; if she can bet on the two at the same frequency (either AIP once per awakening or PIP once per experiment) then she can exploit the fact that they always have the same result at any time during the experiment to make a Dutch book when they have different odds.

And my final thought is that, after halfers and thirders, and Pradeep's position that both are right (but answering different questions), I think it's only fitting that someone supplies the argument "neither position is correct".

Apparently I'm getting to this discussion a week after it ended, but I just chanced upon the site a few days ago (from the link in fivethirtyeight.com's Riddler column this week) and hadn't heard the SB problem before then. I hope I'm not raising a dead discussion, and that this is the right place to post this comment.

• Owen says:

A1Philosopher (and some others) say P(Monday and Heads) = 0.5, while
P(Monday and Tails) = P(Tuesday and Tails) = 0.25.

I guess the Thirders must disagree with these values? However, the values are indisputable if one believes that a head appears with probability 1/2 and that a tail appears with probability 1/2, and that interviews resulting from a tail appearing are equally split between Monday and Tuesday.

On the other hand, P(H) = P(M and Tails) = P(T and Tails) = 1/3. I guess Halvers disagree with these values? However, the values are indisputable if one looks at athe entries of a logbook by the interviewer after thousands of interviews with SB, one entry per interview, where the interviewer wrote down the pair (Day of Week, Coin Status) after each interview.

• Anti-Joker says:

Just heard of this problem, and this seems to have the most recent activity. I believe the answer is unambiguously 1/2, whether you say "the coin landed heads", "the coin shows heads", or "this is a heads awakening". The 1/3 approach, while fine as far as it goes, does not fully model SB's information. The 1/2 approach is the one which takes SB's point of view, as required. On Sunday, she knows the probability is 1/2 each for:

H A_
T AA

Waking up doesn't eliminate either possibility, so when she wakes up her credence for heads is still 1/2. If she knows she will always be told the day, and she hears "Monday", that will also not eliminate either possibility, so her credence for heads is still 1/2.

Now suppose we change the experiment to have two repetitions of the coin toss and questions. On Sunday, SB will know there is a 1/4 probability each for:

H A_ H A_
H A_ T AA
T AA H A_
T AA T AA

When she wakes up, what is the probability that the coin is showing heads? Again, this doesn't eliminate any of the 4 cases. So she must sum the probabilities of heads in each case:

(1/4)(1) + (1/4)(1/3) + (1/4)(1/3) + (1/4)(0) = 5/12.

Interestingly, not 1/2 this time. What if the experiment lasts N weeks? The probability of heads becomes

1/2^N * sum binomial(N,i)i/(2N-i), i=0 to N

Which approaches 1/3 as N goes to infinity.

The probability depends on the number of repetitions! This is the key to the paradox. The 1/2 model not only confirms all our intuitions (the probability of heads is 1/2 in the single repetition case even if you hear "Monday"), but turns out to be capable of using the information SB was given about the number of repetitions to compute surprising probabilities.

Most importantly, as the number of repetitions approaches infinity, the 1/2 model approaches a probability of 1/3 for heads. This eliminates the ambiguity. It has been argued that 1/3 refers to a long run average of wakings, while 1/2 refers to the coin. But the 1/2 model also gives a direct answer of 1/3 with infinite repetitions, so it covers both cases. The 1/3 approach fails because it doesn't fully model SB's Sunday information about the number of repetitions.

Although not necessary to the problem, there are some Monty Hall elements at play that added to the confusion.

Suppose SB is told there will be infinite trials, but every time she wakes up she will be told what week it is. She wakes up and a voice says "week 1". The probability of heads is 1/3! The only way it becomes 1/2 is if she is told on Sunday that there will be only 1 week. That's because she can always be told the week number, just as Monty can always open a losing door. It is essential to model her Sunday knowledge of the number of repetitions, just as it is essential to model Monty's behavior, not just the doors.

A similar issue occurs with the statement "today is Monday." As I showed above, if SB hears "today is Monday" in the standard problem, knowing she will always be told the day, the probability of heads is still 1/2. This is a Monty Hall effect: you must consider the behavior. However, if a day is randomly selected to be named, then Monday is half as likely to be named on tails. In such a case, hearing "today is Monday" actually would raise the probability of heads to 2/3. Not necessary for the problem but clears up some questions.