# Solution: ‘How Many Half-Lives?’

## Introduction

This month’s Insights puzzle was inspired by the negative results of an enormous proton decay experiment in Japan, as Natalie Wolchover recently reported. To help puzzle solvers develop an intuition for what a half-life is, I asked you to answer two simple questions in less than 30 seconds each. These two questions were similar in style to the Cognitive Reflection Test, originated by the decision theorist Shane Frederick. They require you to suppress the seemingly obvious but incorrect answer that first pops into mind and reflect more deeply to arrive at the correct, but less obvious, answer.

## Question 1:

In a bucket filled with nutrients is a bunch of amoebas. Every day, their population doubles. If it takes 48 days for the bucket to be filled with amoebas, how long did it take for the bucket to become half full?

The answer that pops into the mind instantly is half of 48 — 24 days — because we are very familiar with linear process. However, what we are dealing with here is an exponential process. Since the population doubles every day, it had to have been half of the current value the day before, after 47 days. Exponential processes increase or decrease in proportion to their current value. One of the many mystical properties of the base of natural logarithms *e* (2.71828….) that makes it so special is that when you raise it to a variable *x*, the resulting function’s rate of change is exactly equal to its current value — its rise or fall is, in a sense, perfect.

Gerd drew attention to the fact that the illustration should have shown the bucket filled to a different height to make it half full. Since the bucket shown is not cylindrical but has sloping sides, this point is valid and presents a new puzzle, which readers are welcome to attempt (you can assume that the sides of the bucket slope upward at an angle of 100 degrees).

## Question 2:

How many half-lives does a pound of radioactive material have?

This question can be answered from a few different perspectives. Mathematically, the answer is infinite, because there is always a finite probability that some atoms will not have decayed, no matter how much time elapses. From a practical point of view, I said it would be a very large number. Ralf B performed the calculation, and described it as follows (with a correction of the units as pointed out by Rational):

The pound contains 454 grams, or 454*6.022e23 atoms (assuming hydrogen). This is equal to about 2^88 atoms, so they will decay to one in 88 half-lives. For materials other than hydrogen, the number will be slightly smaller.

Ralf B is correct in that this will be the expected number of half-lives. However, decay curves have long tails, so it is quite possible that there will be more half-lives than that. Consider just 2 atoms. A similar calculation says that you should have just 2 half-lives. But each atom has a 1-in-4 chance of not having decayed after 2 half-lives. So the chances that either one is still around after 2 half-lives are pretty good (7/16 or about 44 percent). It would take 8 half-lives to reduce this probability below 1 percent. For 2^{88 }atoms, a calculation using Wolfram Alpha indicates that it would take 95 half-lives to reach a point where the probability that no undecayed atoms exist exceeds 99 percent. So Ralf B’s point is well taken — in practice, the number of half-lives is quite a bit larger than 2, but not quite as large as my words implied. I guess I hadn’t fully absorbed the lesson of Shane Frederick’s Cognitive Reflection test!

If you want to try the other questions from the Cognitive Reflection Test, remember, just 30 seconds per question:

- A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost? _____ cents
- If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets? _____ minutes

Back to half-lives, after the first two simple reflection questions, I confess that I overcompensated by adding two questions that were far too complex for the amount of detail I provided. These questions therefore admitted to many possible answers, as I realized after looking at the various ways readers filled in the gaps. So here, instead of giving a unique solution when there isn’t one, I will discuss interesting insights, assumptions and techniques that readers brought to bear on Questions 3 and 4, all of which have some degree of validity considering the ambiguity in the questions.

## Question 3:

Let us say you manage to obtain 5 atoms of the radioactive isotope of unobtainium, of

Avatarfame. After exactly one year, 2 atoms have decayed. You want to figure out the half-life of the substance, and like a true scientist you seek a range that has a 95 percent probability of containing the true value. What is your range for the plausible half-life of unobtainium?

## Question 4:

In the above scenario, for your second year, you obtain 30 atoms of the substance. You need to divide it into three portions — A, B and C — according to the following rules. When you inspect the portions after exactly one more year, you would like A to have exactly 6 undecayed atoms, B to have 7 or 8, and C to have 4 or 5. Remember, you don’t know the half-life exactly. How many atoms must be in A, B and C initially in order to maximize your chances of getting the precise results you want?

The simplest approach to these questions was adopted by Gerd, Luca U. and Tzara Duchamp, who calculated on the basis of the sample measurement alone to obtain an exponential decay constant of ln (5/3), giving a half-life of 1.36 years. For Question 4, the best way to split the groups then turns out to be 10, 13 and 7 atoms.

However, as John Pickering pointed out:

The way the question is phrased (seek a range that has a 95 percent probability of containing the true value) means that what is required is a Bayesian posterior (or credible) interval rather than simply a confidence interval (meaning that if the experiment was repeated multiple times the confidence interval calculated would contain the true value 95 percent of the time).

John’s point is worthy of deep consideration. When we make a statistical observation of a variable like half-life or height by examining a sample population, the measurement by itself cannot tell you the true value of the variable within the population. By random variation, our sample mean might be higher or lower than the true mean. But statisticians can calculate a 95 percent “confidence interval” as John explained: If the experiment were repeated multiple times, the confidence interval calculated would contain the true value 95 percent of the time. This can be done entirely based on the measurements from the samples. Each time you did the experiment, you would get a different confidence interval, but if you did it a hundred times, the true half-life or height would be within your stated intervals about 95 percent of the time.

However, this is subtly different from finding a range that has a 95 percent probability of containing the true value: In this case we cannot just rely on measurements from a sample, but we need some real world knowledge about how the true value is distributed (a Bayesian prior) in order to calculate the probability distribution of which we can exclude tails on the left and right sides representing a total probability of 5 percent. In the absence of clear direction in the questions on these points, readers came up with some ingenious extrapolations of their own. One possible prior assumption could have been to assume an equal chance of the half-life for equal periods of time (there is an equal chance of the half-life in the range 0-1 years as there is over 1-2 years), or that the decay rate is uniform, as Mark P assumed. The problem is that this obviously cannot be true for all the infinite periods of time that exist, or else the probability cannot add up to one. After some arbitrary point in time you have to assume that the chances of the half-life taper off to zero. Where that point is located, at, say, 5, 10 or 20 years, will affect the values of the true half-life and change the answers.

Nevertheless, several readers, such as John Pickering, Mark, Mark P and Joseph Fine, sketched out the general method very well: Figure out the probability distribution and then lop off enough of the left and right tails to leave 95 percent of the probability in the center. Here I liked Mark’s approach of not taking 2.5 percent of the probability distribution from the left and right tails (since they are asymmetric), but rather find a probability density that is equal on the left and right sides and encloses 95 percent of the probability distribution function. Very nice! Unfortunately, because of the differences in their assumptions, for which I take full responsibility, all the above readers obtained different ranges: Jon Pickering got 68 days to 27 years; Mark obtained 0.484 to 6.26 years; Mark P’s range was 0.9 to 6.3 years and Joseph Fine’s 0.755 to 3.106 years. All of you are winners. With such a small sample, a frequentist approach that might have given an accurate answer is untenable.

I find Mark P’s following speculation intriguing:

I kind of think that the obvious division of 10, 12, 8 or 10, 13, 7 is misguided. Those answers assume that the decay rate is exactly 0.4 per year. If it’s much higher or much lower than that — which is reasonable given the tiny sample size — using 10, 12, 8 or 10, 13, 7 is likely to overshoot or undershoot the mark by a wide margin.

That was the kind of situation I was trying to set up. Unfortunately, the different models will all yield different answers. In retrospect, I missed a chance to ask a far clearer question more relevant to the negative proton decay result. Here is my do-over, for those who may still have the stamina to think about it. The answer is equal parts mathematical and psychological, just like the proton decay result.

## Half-Life Do-Over Question:

Let us say you manage to obtain 5 atoms of the radioactive isotope of unobtainium, of

Avatarfame. On theoretical grounds you have determined that its half-life is exactly one year. After exactly one year, no atoms have decayed. After how many years of obtaining this negative result will it be less than 5 percent likely that your theory is true? When will you be ready to admit that your theory is dead?

Thank you to all who attempted these ambiguous questions. I’m happy that some of you still had fun with it and contributed insights worth sharing.

The *Quanta* T-shirt for this month is still up for grabs. Do submit your contributions here.

*Updated on February 10, 2017*

The comments below have raised interesting issues:

**Binomial vs. Poisson:**Ashish used the binomial distribution in his calculation for the do-over question, whereas Joseph Fine used the Poisson distribution. Which one applies here? The Poisson distribution is an easy approximation of the binomial distribution that’s useful when the time interval of interest is a small fraction of the half-life — instances in which binomial calculations quickly become intractable. For this reason the Poisson distribution is the one used in serious scientific calculations such as the proton decay experiment. However, in the case of our question, the decay time is very short — on the order of the observation time — and therefore the binomial distribution, which is more accurate, should be used.

**Truth and Probability:**Ethaniel made a point about tightening the language regarding probability and the truth of a given theory, which is well taken. As he pointed out, the right question to answer is, “If you assume that your theory is true, how long does it take for the no-decay observation to fall under the 5 percent likelihood threshold?” The answer for the do-over question is 0.864 years, as determined by Ashish, and for the die-hards who want the likelihood to fall under the 5-sigma threshold (1 in 3.5 million, much used in physics experiments such as the proton decay calculation), it is 4.35 years, as determined by Ethaniel.

**Letting Go of a Theory:**Will some die-hards cling to their pet theories even longer than that? Mark explored the psychological nature of the question “When will you be ready to admit that your theory is dead?” by carrying out a number of elegant investigations that show how the strength of the initial belief affects how long you are willing to wait before losing hope. There are two famous anecdotes regarding this point.

The first concerns Albert Einstein, who in 1906 was confronted with an experimental result that seemed to go against his special theory of relativity while supporting some rival theories. According to his biographer Banesh Hoffmann, Einstein looked at the rival theories with “aesthetic disapproval” and confidently suggested that the experimenter might be mistaken. As we now know, Einstein’s self-belief was vindicated.

The second is a famous quote by Max Planck: “Science advances one funeral at a time.” Even in science, some die-hards never give up their pet theories. Sadly, unlike the other occasion, this was Einstein’s fate with quantum mechanics. It’s the new generation that embraces and advances new results.

For showing this mathematically, and for his earlier neat method capturing 95 percent of the probability distribution mentioned above, the *Quanta* T-shirt for this Insights puzzle goes to Mark. Congratulations!