You survey many many school children and ask each one:

Do you have more brothers than sisters? or more sisters than brothers? or the same number?

Then you separate the boys’s answers from the girls’s answers

Which of the following is true?

- The answers from boys will be tilted toward having MORE sisters and LESS brothers
- The answers from boys will be tilted toward having MORE brothers and LESS sisters
- There will be no (statistically significant) difference.

**The reason for answer 1 is**: Since the ratio of boys/girls is 50/50, and you don’t count yourself in the answer, boys will tend to have more sisters and girls will tend to have more brothers

**The reason for answer 2 **is: Some families are uneven and have more boys. In these families you will see more boys having more brothers than sisters and less girls having more brothers than sisters. Some families are uneven and have more girls. In these families you will see more girls having more sisters than brothers and less boys having more sisters than brothers.

I thank Oded Margalit for this question given to me as a birthday gift.

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Here is another model that favors 2. If parents stops giving birth after they collect both sexes, the number of boys brothers will be something like \sum n/ 2^n, which is slightly larger than 1/2? So I suspect the answer depends on the birth model.

Almost certainly depends on the birth model! Should we assume family size is independent

of numbers of boys/girls, and that boys and girls are equally likely (in each family)? Can we first make the puzzle well-posed?

In the best Jewish tradition I’ll answer a question with another question – one I’ve asked my students and colleagues and the record ain’t very good.

In country X the government enacted the law that every family is allowed only one son and, once you have a son, the family must stop having any extra children. The boundary conditions are (1) 50/50 ratio, (2) all children born to steady, life-long families, (3) totally effective enforcement: no foetuses aborted because of their sex, neither are babies, and, to make it even more unrealistic, (4) there is no corruption and everybody (yes, even Comrade Chairman) follows the rules. The question: up to roundoff error, what is the proportion of female and male children?

A version of this problem generated a flame war on MathOverflow. https://mathoverflow.net/a/17963

A variant I thought was interesting: If the survey also records how many more brothers than sisters they have (or vice versa), and we look at the answers given by all the boys, will the average excess be (statistically) positive/negative/zero ?

One way to think about the problem is this: Suppose you draw at random 500 school children from NYC and check for each one which of the three alternatives hold (more brothers, more sisters, same number). Would you expect (statistically significant) difference and in which direction?

Another way (which is probably the original intention) is to make the strongest simplifying assumptions. The reasons in the post for the opposite outcomes apply even there, so what is the correct answer?

And then, following Nick’s comment another challenge is to find the simplest birth model for which the answer changes, if there is any.

Ah, I came here hoping for a thought-provoking probability riddle and that is just what I found. Thank you, Gil!

If we posit that parents’ decision to have more children (not always a decision, I know, so let us say their decisions regarding behaviors that lead to more children) does not depend on the sex of their existing children, then I believe the answer is #3 (and the reasons given for #1 and #2 are red herrings). In this scenario, if we pick a child at random, their number of brothers and number of sisters are identically distributed, and their own sex is not relevant.

Of course birth models that are not sex-blind in this way need not produce such a result. In Arieh’s birth model, some boys have more sisters than brothers, but none have more brothers than sisters.

While thinking about this problem, I noticed something utterly elementary that I cannot remember noticing before. One reasonable model for family size is geometric: this results from positing that once a family has k children, their probability of having additional children is independent of k. If we additionally assume that each birth is equally likely to produce a girl or a boy, (or just that the proportion of girls to boys is independent of birth order,) then the number of boys in a given family is itself geometrically distributed, and so is the number of girls. However, the number of boys and the number of girls are not independent. (If they were, then family size would be negative-binomial.) I can get these conclusions to comport with my intuition, but only after reasoning with my intuition. Maybe it’s not intuition if you have to reason with it.