The older child is a boy, so are both boys?

Question

A family has two children, each independently a boy or a girl with probability $1/2$ . This time you learn something more specific: the older child is a boy.

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

Accepted Answer

Write the older child first in each equally likely pairing: \{BB,\ BG,\ GB,\ GG\}. "The older child is a boy" keeps only the cases whose first slot is B: \{BB,\ BG\}. Of these two, exactly one is all boys: P(	ext{both boys} \mid 	ext{older is a boy}) = \frac{1}{2}. Compare this with the vaguer clue "at least one boy," which gives 1/3. Naming a specific child (the older one) is stronger information: it fixes one slot, so the only uncertainty left is the single younger child, a clean coin flip. Takeaway. A clue that pins down a specific individual conditions more sharply than "at least one," …

The older child is a boy, so are both boys?

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