Do men or women have more brothers?
I think women have more as no man can be his own brother. But how one can prove it rigorously?
I am going to suggest some reasonable background assumptions:
- There are a large number of individuals, of whom half are men and half are women.
- The individuals are partitioned into nonempty families.
- The distribution of the sizes of the families is deliberately not specified.
- However, in each family, the sex of each member is independent of the sexes of the other members.
I believe these assumptions are roughly correct for the world we actually live in.
Even in the absence of any information about point 3, what can one say about relative expectation of the random variables “Number of brothers of individual I, given that I is female” and “Number of brothers of individual I, given that I is male”?
And how can one directly refute the argument that claims that the second expectation should almost certainly be smaller than the first, based on the observation that in any single family, say with two girls and one boy, the girls have at least as many brothers as do the boys, and usually more.
So many long answers! But really it’s quite simple.
- Mathematically, the expected number of brothers is the same for men and women.
- In real life, we can expect men to have slightly more brothers than women.
Assume, as the question puts it, that “in each family, the sex of each member is independent of the sexes of the other members”. This is all we assume: we don’t get to pick a particular set of families. (This is essential: If we were to choose the collection of families we consider, we can find collections where the men have more brothers, collections where the women have more brothers, or where the numbers are equal: we can get the answer to come out any way at all.)
I’ll write p for the gender ratio, i.e. the proportion of all people who are men. In real life p is close to 0.5, but this doesn’t make any difference. In any random set of n persons, the expected (average) number of men is n⋅p.
- Take an arbitrary child x, and let n be the number of children in x‘s family.
- Let S(x) be the set of x‘s siblings. Note that there are no gender-related restrictions on S(x): It’s just the set of children other than x.
- Obviously, the expected number of x‘s brothers is the expected number of men in S(x).
- So what is the expected number of men in this set? Since x has n−1 siblings, it’s just (n−1)⋅p, or approximately (n−1)÷2, regardless of x‘s gender. That’s all there is to it.
Note that the gender of x didn’t figure in this calculation at all. If we were to choose an arbitrary boy or an arbitrary girl in step 1, the calculation would be exactly the same, since S(x) is not dependent on x‘s gender.
In real life:
In reality, the gender distribution of children does depend on the parents a little bit (for biological reasons that are beyond the scope of math.se). I.e., the distribution of genders in families is not completely random. Suppose some couples cannot have boys, some might be unable to have girls, etc. In such a case, being male is evidence that your parents can have a boy, which (very) slightly raises the odds that you can have a brother.
In other words: If the likelihood of having boys does depend on the family, men on average have more brothers, not fewer. (I am expressly putting aside the “family planning” scenario where people choose to have more children depending on the gender of the ones they have. If you allow this, anything could happen.)