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A Surprising Probability Result

At dinner last night, several of us were discussing the Chinese one-child per family policy, when Sandy Blank posed the following question.

Suppose the probability that a child is male is exactly 1/2, and that each couple continues to have children until a male is produced, and then stops. What fraction of the new generation will be male?

Upon hearing this question, most everyone will guess that the number of girls will be considerably greater than the number of boys.

I reasoned that each completed family will have one boy, and that I could compute the expected value of the number of girls by summing n x p(n) for all positive integers n, where p(n) is the probability that the couple will have n consecutive girls before having a boy. See here for the details of the computation, which are relatively simple.

I was shocked to find that the expected value of the number of girls was also one, so the new generation will be 1/2 male.

My sister Arlene, a statistician, was well aware of this problem, and she presented an incredibly simple solution. Since each birth has a probability of 1/2 of being male, the new generation will be approximately 1/2 male. It doesn't matter when families decide to stop having children.

I think that (with either solution) this is a neat problem. It might also make a good bar bet.

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