If every family only has one child, nobody has any brothers; the results are equal at 0 average brothers. If we take some distribution over size n of the famil

If every family only has one child, nobody has any brothers; the results are equal at 0 average brothers. If we take some distribution over size n of the families, say a distribution where family size is either 1 or 2, then the 1-size families don't contribute (they add a zero to the fraction, dragging the probability down in a systematic way equal for both sexes), and the 2-size families have four equally probable outcomes. those four outcomes (bb bg gb gg) yield: two females with no brothers. two females with one brother. two males with no brothers. two males with one brother. i.e. the average numbers of brothers are equal (1/2 for males, and the same for females.) The next layer, where n = 3, there are (ggg ggb gbg bgg gbb bgb bbg bbb) 8 combinations, yielding: three females with no brothers. six females with one brother. three females with two brothers. three males with no brothers. six males with one brother. three males with two brothers. Average number of brothers for females is (30 + 61 + 32) / 12 = 1. Average number of brothers for males is (32 + 61 + 30) / 12 = 1. for n = 4, (gggg gggb ggbg gbgg bggg ggbb gbgb gbbg bggb bgbg bbgg gbbb bgbb bbgb bbbg bbbb), there are 16 combinations (yes, these map to binary integers quite nicely): four females with no brothers. twelve females with one brother. twelve females with two brothers. four females with three brothers. four males with no brothers. twelve males with one brother. twelve males with two brothers. four males with three brothers. Again, this comes out at an even probability - 48 / 32 = 2 brothers, on average. By induction, we can say that for any n, the probabilities are equal (if the probability of male and female children is equal). Thus, for any distribution of family sizes, the probabilities are equal. --- Real world criticisms: * males are more likely (~51% of births naturally, higher thanks to sex-selective terminations in China and India.) Thus, outcomes with more males are more likely, which skews towards male-heavy families. While this means both males and females have more brothers, there are more males to count with higher numbers of brothers, so their average goes higher. * males die more frequently at younger ages, so older people will have fewer brothers than younger people.