I was going to wait a few days before posting the answer to yesterday’s puzzler, but we’re up well over 100 comments already, the holidays are almost upon us, and I think it’s time to settle this so you can all give your full attention to whatever festivities you’ve got coming up.
Here’s the puzzle again:
More precisely: What fraction of the population should we expect to be female? That is, in a large number of similar countries, what would be the average proportion of females?
Stop reading here if you don’t want spoilers:
- Here’s the wrong answer: Every birth has a 50% chance of producing a girl. This remains the case no matter what stopping rule the parents are using. Therefore the expected number of girls is equal to the expected number of boys. So in expectation, half of all children are girls.
- Pretty convincing, eh? So why is it wrong? Well, actually, most of it is right. Every birth has a 50% chance of producing a girl — check. This remains the case no matter what stopping rule the parents are using — check. Therefore the expected number of girls is equal to the expected number of boys — check! But it does not follow that in expectation, half of all children are girls!
- To see why not, let me tell you about the families who live on my block. There are 3 families with four girls each (and no boys), and one family with 12 boys (and no girls). Altogether, that makes 12 girls and 12 boys — equal numbers! On average, each family has three girls and three boys. Nevertheless, the fraction of girls in the average family is not 50%. It’s 75% (the average of 100%, 100%, 100%, and 0%).
- In other words, if you were to choose a random family off my block, the expected number of girls would equal the expected number of boys — 3 in either case. But the expected fraction of girls in the family would be 75%. Moral: Just because two variables have an expected difference of zero, you can’t conclude they have an expected ratio of one. That needs to be computed separately.
Edit: This came up in comments, and it might be worth adding here: This example in no way relies on there actually being four families. Suppose there’s just one family, that randomly decides whether to adopt four girls (with probability 75%) or twelve boys (with probability 25%). In that family, the expected number of girls equals the expected number of boys, and the expected fraction of girls is still 75%. The country in the original problem is drawn randomly from a universe of possible countries, just as this family is drawn randomly from a universe of possible families.
So that explains why the “obvious” argument is wrong, and that, to me, is really the interesting part. But of course we’re not done until we find the right argument. That’s a bit trickier, and it depends on the country’s population. I’ll start with the case where there’s just one couple. Here are some possible family configurations, with their probabilities:
From this we see that the expected number of boys is
which adds to 1. And the expected number of girls is
which also adds to 1. Sure enough, the expected number of girls is equal to the expected number of boys.
But the expected fraction of girls is
which adds to 1-log(2), or about 30.6%.
For a population of k families, a similar calculation gives an answer of approximately (but not exactly) (1/2) – (1/4k), which, when k is large, is approximately (but not exactly) 1/2.
Several of our commenters were on to various aspects of this, and some were on to pretty much all of it. In no particular order let me acknowledge Vic, DaveB, KenB, ThomasBayes, loveactuary, JonathanCampbell, Brett, wellplacedadjective, JonathanKariv, and mobile — and let me apologize for anyone I’ve inadvertently omitted (with so many comments to digest, I’m sure there are a few). But above all, a humble tip of the hat to the mathematician and backgammon expert Douglas Zare who inspired this post with his brilliant exposition over at MathOverflow (his is the first of the several answers). (Warning: Depending on your technical background, you will find his explanation either perfectly illuminating, perfectly indecipherable, or somewhere in between.)
Now go enjoy your holiday. If your relatives like this kind of thing, you can share it with them. If not, you can use it to get them to leave you alone.