A Probability Challenge

Last week I had the pleasure of meeting Alex Bellos in Oxford. Among other things Bellos writes the Guardian Monday puzzle column. He gave me a copy of his latest book, Puzzle Me Twice, where the obvious answer is not correct. I got more right than wrong, but I hated being wrong. Here is one of those puzzles, Sistery Mystery (page 28), which is a variation of a puzzle from Rob Eastaway. 

Puzzle 1: Suppose the probability of a girl is 51% independently and uniformly over all children. In expectation, who has more sisters, a boy or a girl?

Go ahead and try to solve this before reading further.

In any family with both boys and girls, each boy will have one more sister than each girl. For example in a family with four girls, each boy has four sisters and each girl only has three. Thus boys have more sisters on average.

Wrong, it's exactly the same. To see this consider Alex, the sixth child of a ten-child family. The number of Alex's sisters is independent of Alex's gender. This is a pretty robust result, it doesn't depend on the probability of a child being a girl, or if we allow non-binary children, or if the distributions aren't identical, say the probability of a girl is higher for later kids in a family. All you need is independence.

So what's wrong with my initial intuition that in every family boys have more sisters than girls. Eastaway suggests this gets balanced by the families of a single gender, but this happens rarely for large families. Instead it's a variation of Simpson's paradox. The naive argument doesn't account for the fact that girls are overrepresented in girl-heavy families. Consider a family of two boys and eight girls. Each of the two boys has eight sisters but four times as many girls have seven sisters, which adds to the expected value more to the girls than the boys.

If you lose independence the solution may not hold, for example if we have identical twins. 

I'll leave you with one more puzzle.

Puzzle 2: Suppose you are in a country where each family has children until they get their first boy. In this country, do boys or girls have more sisters on average?

Answer below.

In Puzzle 2 we lose independence and if Alex is a girl, she's more likely to be in a family with many girls. Indeed if boys and girls have equal probability, when you work out the infinite sums in expectation a boy will have one sister and a girl will have two.