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Boy ratio when families stop at the first son

Question

In some society each couple keeps having children until a son is born, and then stops. Every birth is independently a boy or a girl with probability 12\tfrac12 each. As the population grows over time, what fraction of the children born are boys?

Short answer

50%. The stopping rule changes how many children each family has, but every individual birth is still an independent fair coin flip. Hence the expected numbers of boys and girls born are always equal, and the proportion of boys tends to one half.

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