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.
I had the opposite intuition, because there are more sisters of girls than there are sisters of boys.
ReplyDelete"The number of Alex's sisters is independent of Alex's gender."
ReplyDeleteBut in the same family, a male Alex has one more sister than a female Alex.
In Puzzle #1, the 51% is a red herring, isn't it?
ReplyDeleteVariants/formalizations of puzzle #1:
ReplyDelete(a) Pick a boy at random. Pick a girl at random. Which has a higher expected number of sisters?
(b) Pick a child at random. Conditioned on picking a boy versus conditioned on picking a girl, which has more expected sisters?
(c) Pick a child at random and pick a sibling at random. Which is a higher probability, picking a boy and his sister, or picking a girl and her sister?
@E: Should hold for 0<p<1; while independence technically doesn't break at the remaining two values that p could theoretically take, i.e., p=0 or p=1; with probability 1, the question would "break down."
ReplyDelete@Lance: What puzzled me and then amused with probability 1 was February 26th's Wordle Word. It might amuse you too ...
"The number of Alex's sisters is independent of Alex's gender."
ReplyDeleteRight. If you remove Alex from the sample pool, then Alex is looking at 9 siblings randomly chosen from (49% boys + 51% girls) and sees exactly a 51% probability count of sisters.
But, since this is a complexity blog, and in complexity, we do combinatorics, that's friggin' got to be wrong.
In all possible combinations of boys + girls, the girls always see one less sister than any boy does. One girl = Zero sisters, all girls = n - 1 sisters. Zero girls, is the only case where the girls don't get dinged for self-sisterhood not being allowed. And zero boys is the only case where boys don't get an extra sister (but the girls still get dinged!).
And don't forget the zero boys AND zero girls families!
Ouch. Combinatorics is hard. Still, I think the probabilist argument is simply wrong.
some commenter said probabilities don't matter. you say they do?
DeleteMy "right" was a sarcastic introduction to (my explanation of) the given solution to problem 1, which I think is wrong, but I haven't succeeded in working out the details thereof.
DeleteDitto for problem 2. For every boy with one sister, there's one woman with no sisters. For every boy with two sisters, there are two sisters each with one sister.
Thus there are two similar infinite (but converging in real life) sums. 50% of boys have no sisters, 25% have one, etc.. For a first daughter, she has a 50% chance of being an only daughter (second kid was a boy). So 50% of women have no sisters, and 25% have one sister, etc.
So here, the combinatorics is clearer (to me, anyway): the number of sisters is the same for boys and girls, despite the obnoxious sexism of the society. I think.
I reserve the right to be completely dead wrong.
Something's wrong here. P1 makes no assumptions about parents' stopping strategy, so how could P2 give a different answer?
ReplyDelete