At least one boy, but are both boys?

Question

A family has two children. Each child is independently a boy or a girl with probability $1/2$ . You learn only that at least one of the two children is a boy.

Given that, what is the probability that both children are boys?

Accepted Answer

List every equally likely pairing, writing the older child first: \{BB,\ BG,\ GB,\ GG\}. The clue "at least one boy" rules out GG, leaving three equally likely cases: \{BB,\ BG,\ GB\}. Only one of these three, namely BB, has both boys: P(	ext{both boys} \mid 	ext{at least one boy}) = \frac{1}{3}. The surprise is that the answer is not 1/2. Knowing "at least one boy" does not point to a specific child, so it keeps the two mixed families BG and GB alive, and they outnumber the all-boy family. Takeaway. Conditioning shrinks the sample space to the outcomes consistent with your clue; vague info…

At least one boy, but are both boys?

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