From Air Canada’s inflight magazine:

And since I’m sure someone will ask, here is, apparently, the only solution they could think of:

Continue reading ‘Worst Puzzle Ever’

A man goes to the doctor, gets a prescription for a headache medicine, and dies the next day. It’s known that 60% of those who receive these prescriptions actually fill them and take the medicine. It’s also been established by investigators that if the man took the medicine, then there’s a 90% chance it killed him. What’s the probability that he took the medicine and it killed him?

The answer might depend on your auxiliary assumptions, but there is a particularly simple and natural set of auxiliary assumptions that leads to a nice clean answer. And no, that answer is not 54%. (Nor would 54% be an easy answer to defend under any reasonable assumptions I can think of.)

EDIT: I had written here For extra clarity, the phrase “the medicine killed him” should be interpreted to mean that if he hadn’t taken the medicine, he wouldn’t have died. . This seems to be confusing some readers, and I briefly posted an edit here saying to ignore it — but it actually is what I meant to say all along.

Continue reading ‘Another Probability Puzzle’

Last week’s series of posts on boys, girls and population ratios drew an astonishing 850 or more comments, and they’re still coming in. There’s a lot of stuff worth reading in those comments, but the conversation — which is spread out over four threads and often involves several overlapping subconversations — has become difficult to follow. Fortunately, one of our more insightful commenters has volunteered to write a guest post summarizing what he sees as some of the most important discoveries to come out of those discussions. He has signed his guest post “Tom M”, but in the comments, he is simply “Tom”.

Without further ado:

1. Introduction

Steve Landsburg posed a problem in this post at his blog entitled Are You Smarter than Google?. Here was the problem statement:

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Well, of course, you can’t know for sure, because, by some extraordinary coincidence, the last 100,000 families in a row might have gotten boys on the first try. But in expectation, what fraction of the population is female? In other words, if there were many such countries, what fraction would you expect to observe on average?

As we’ve worked on that problem, one of the points that came up was something called the “extra half boy.”

In this post I’m going to try to summarize what we’ve said about that critter and some related issues.

A key reference, that everybody should read, is What is the expected proportion of girls?, by ‘Thomas Bayes’.

Continue reading ‘The Extra Half Boy’

As promised, I am providing an update on the status of the various bets that got placed here last week.

The problem is this: In a certain country, each family continues having children until it has a boy, then stops. In expectation, what fraction of the population is female?

The answer is: not 1/2. A more precise answer depends on your additional assumptions — the number of families, the mortality rate, whether the parents count, whether there are grandchildren, etc. It might be possible to engineer those additional assumptions in some way that makes the answer come out to 1/2, but it would surely be impossible to argue that those ad hoc assumptions, whatever they are, are implicit in the statement of the problem.

Pretty much everyone seems to understand this now. (Here’s one more learning aid, if you’re still struggling.) What, then, are we betting about? Here is a list of all the open and closed bets:

Continue reading ‘Status of the Bets’

There’s a certain country where everybody wants to have a son. Therefore each couple keeps having children until they have a boy; then they stop. What fraction of the population is female?

Well, of course, you can’t know for sure, because, by some extraordinary coincidence, the last 100,000 families in a row might have gotten boys on the first try. But in expectation, what fraction of the population is female? In other words, if there were many such countries, what fraction would you expect to observe on average?

I first heard this problem decades ago, and so, perhaps, did you. It comes up in job interviews at places like Google. The answer they expect is simple, definitive and wrong.

Continue reading ‘Are You Smarter Than Google?’

Many years ago, when I was teaching at Cornell, my then-colleague Nick Kiefer kept me endlessly amused with the creative assignments he dreamed up for the students in his Decision Theory class. I’m reporting this from memory, so I don’t guarantee that I have it exactly right, but I think this was one of those problems:

• On Day Zero (before classes start), Nick uses his pocket calculator to generate, at random, a secret number between 0 and 100. This number is drawn from a uniform distribution, which means (roughly) that the drawing is unbiased, so any number is as likely to come up as any other. The students’ job is to guess this number.
• On Day One — the first day of class — Nick uses his pocket calculator to generate a Number of the Day. He doesn’t tell the students what the Number of the Day is, but he does tell them whether it’s larger or smaller than the secret number.
• On Days Two, Three, and so forth, he does the same thing. He generates a Number of the Day and announces whether it’s bigger or smaller than the secret number.
• You, the student, can submit your guess on the day of your choice.
• You are then assessed two penalties. The first penalty is equal to the square of the error in your guess. So if the correct number is 3 and your guess is 5, your penalty is 2², or 4. The second penalty is equal to the natural logarithm of the day when you submit your guess. So if you submit your guess on Day 7, your penalty is the log of 7, which is 1.94591.
• These penalties are subtracted from some some fixed number of points that you’re given to start out with (say 1000). The result, after the subtraction, is your score for the assignment. This score counts for a significant fraction of your course grade.

Continue reading ‘How to Decide’

Here I have a couple of urns. The one on the left contains 70 red balls and 30 black. The one on the right contains 30 red and 70 black.

While you weren’t looking, I reached into one of these urns and randomly drew out a dozen balls. As you can see, 4 of them were red and 8 were black.

Here are three questions that I think you ought to be able to answer if you want to be in the business of assessing evidence:

1. If you had to guess, which urn would you guess I drew from?
2. What’s your estimate of the odds that you’re right?
3. Do you think you’re right beyond a reasonable doubt?

Further discussion to follow later in the week. (Hat tip to Howard Raiffa, who will also figure in the upcoming discussion.)