Brainteasers & Puzzles

All-Girl World?

Every couple has children until a girl arrives, then stops. Does the stopping rule skew the fraction of girls in the world?

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In a society every couple keeps having children until they get a girl, then stops. Each child is a girl with probability one half, independently of the others. Over time, what happens to the fraction of girls in the society?

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#Count one family

Every family ends on its first girl, so it has exactly one girl. Before her come some number of boys, each an independent flip that happened to land boy. The chance of exactly kk boys is (12)k+1\left(\tfrac{1}{2}\right)^{k+1}, and the expected number of boys is

E[boys]=∑k=0∞k(12)k+1=1.(1)\E[\text{boys}] = \sum_{k=0}^{\infty} k\left(\tfrac{1}{2}\right)^{k+1} = 1. \tag{1}

So a family carries one girl and, on average, one boy.

#The whole society

Summing over many families, the children split evenly, about one girl and one boy each, so the fraction of girls tends to

E[girls]E[children]=11+1=12.(2)\frac{\E[\text{girls}]}{\E[\text{children}]} = \frac{1}{1 + 1} = \frac{1}{2}. \tag{2}
Each family stops at its first girl, so it adds one girl and, on average, one boy. Across 8,000 families the share of girls holds at one half however many boys come first.

#Why the rule changes nothing

The stopping rule decides when a family stops, not how any single birth lands. Each birth is still a fair coin, and no rule that looks only at past births can tilt the next one. Boys pile up in the long families and are absent from the short ones, yet across the society they balance the girls exactly.