Do boys have more sisters than girls do?

One of the famous intuitive mistakes in probability comes from the simple question, "Do boys have more sisters than girls do?" A quick analysis of the situation may prompt you to say yes. A more in-depth look might change your mind.

I don't have a sister. I have to say, I don't feel much deprived, except when I think of the millions I might make if I were to get my sister to drive across America with me, or volunteer in Bangladesh with me, or meditate with me every morning for a year, and then write a sappy best-selling memoir about it. The question is, did being born female decrease my odds of having a sister?

This is the set-up for a well-known question in probability. A quick look at the idea might give you the impression that, yes, boys have more sisters than girls do. A two-child family that includes a boy and a girl gives the boy a sister, and the girl a nothing but a contemptible brother. Even if the parents had another child, and provided their daughter with a sister, the boy would now have two sisters to the girl's one. Boys have the advantage, it seems.

Or do they? Let's look at the cases one by one. A two-child family can have four different combinations, two girls, an elder girl and a younger boy, an elder boy and a younger girl, and two boys. The families with two children of the same sex provide no sisters to any boy, but one sister to each of the two girls - making for two girls with two sisters. The families with one boy and one girl provide no sisters for the girls, but one sister for each of the two boys - two boys with two sisters. So far each sex is provided with equal amounts of sisters.

How about three-child families? There are three general combinations. A family of three boys yields zero sisters for anyone. A family of three girls nets three separate girls two sisters each, making for six sisters for three girls. So far, six sisters for female children.

A family with two boys and one girl gives each boy a sister, making for two sisters. A family with two girls and one boy provides two sisters for the boy, and one sister for each of the girls. In total that's four sisters for boys, and two sisters for girls. There is one final trick, though. Each of those families comes in one of three different combinations. For example, the boy-boy-girl family could have the girl be the eldest, middle, or youngest child. So we have to triple the figures we got from both of these examples - making twelve sister for boys, and six sisters for girls.

Add all the three-child family possibilities together and you get twelve sisters for girls, and twelve sisters for boys. There's no way around it. Girls have the same number of sisters that boys do.

Now, if you'll excuse me, I'm going to have to go convince my brother to help me write a memoir.

[Via Magnificent Mistakes in Mathematics.]