Today I have a new brain teaser for you. It’s similar in flavor to last week’s brain teaser, but it’s genuinely different, and it’s different in an instructive way. And it stands on its own; you don’t need to have followed last week’s discussion to tackle this.
Here goes: 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. In expectation, what is the ratio of boys to girls?
For those who were here last week, notice that this problem is genuinely different. Last week I asked about the fraction of the population that is female. If we exclude the parents, that’s the ratio G/G+B. Today’s problem asks about the ratio B/G.
Stop here if you don’t want spoilers.
Here’s the wrong argument: Each birth is equally likely to be a boy or a girl, so we should expect equal numbers of boys and girls. Therefore the expected ratio of boys to girls is 1. This argument is wrong for exactly the same reason that 1/2 was the wrong answer to last week’s problem — it’s true that the expected difference between girls and boys is zero, but that tells us nothing about the expected ratio.
The right answer is not 1. It’s infinity.
That’s because there’s always some (possibly tiny) chance that zero girls have been born, making the ratio infinite. The expected ratio is an average over all possible outcomes. When one of the things you’re averaging over is infinite, then the average is infinite.
What’s interesting about this answer is that it is nowhere close to the “natural” answer of 1. So the erroneous reasoning that gets you to that natural answer must not be even remotely trustworthy.
There’s a lesson here. To see what the lesson is, you have to know what happened last week:
- I posed a problem.
- Many readers gave incorrect arguments, concluding that the answer was 1/2.
- The correct answer, however, was not 1/2, though it was close to 1/2 for a large country.
- Many of those readers then said “Ha! So my answer was right after all — or at least so close to right that it ought to count! To call my answer wrong is to quibble over an insignificant difference!”
- I said, “No, your reasoning is completely wrong. If it got you close to the right answer, that’s the kind of fluke that gets no credit.”
- Many readers responded: “But since my answer is so close to right, my reasoning must be essentially valid”.
- I said: “Huh?
The nice thing about today’s puzzle is that it illustrates why those readers have no case. Because exactly the same incorrect reasoning that gave you the answer 1/2 last week will give you the answer 1 this week. But this time, your answer is nowhere close to right.
Let me say that again: If you stick to the reasoning that so many readers defended so vehemently last week, then this week you will get an answer that is off by infinity. That’s a pretty good reason to reject that kind of reasoning.