The Two-Child Problem

Question

A family has two children. You learn that at least one of them is a boy. What is the probability that both are boys? (Assume boys and girls are equally likely and independent.)

Accepted Answer

The four equally likely orderings are BB, BG, GB, GG. The information "at least one boy" eliminates only GG, leaving three equally likely outcomes, of which one is BB: P(BB mid textat least one boy) = frac1/43/4 = frac13. The common wrong answer, frac12, silently answers a different question, "the first/specific child is a boy", which eliminates two cells and leaves BB, BG. The famous follow-up "At least one is a boy born on a Tuesday" shifts the answer to frac1327. The more specific the identifying information, the closer you get to having identified a particular child, and the closer the answer drifts to frac12. The lesson interviewers want: how you learned information changes what it tells you.

The Two-Child Problem

Problem

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