the two-child (boy-girl) problem
A family has two children. The puzzle compares two seemingly similar questions whose answers differ, exposing how much the exact wording of the information matters. Question one: given that at least one child is a boy, what is the probability both are boys? Question two: given that the older child is a boy, what is the probability both are boys? The surprise is that these are 1/3 and 1/2 respectively, not the same.
Start from the four equally likely birth orders: BB, BG, GB, GG (older listed first). For question one, 'at least one boy' removes only GG, leaving BB, BG, GB — three worlds, of which one (BB) has two boys, so the answer is 1/3. For question two, 'the older is a boy' removes GG and GB, leaving BB and BG — two worlds, of which one (BB) has two boys, so the answer is 1/2. The difference is entirely in what the condition rules out: naming a specific child (the older) collapses the space more than the vaguer 'at least one'.
The deeper lesson is that the answer depends on HOW you came to know there is a boy — the sampling procedure, not just the bare fact. A famous variant adds a detail: 'at least one boy born on a Tuesday' shifts the answer to about 13/27, which startles people until they recount the conditioned outcomes. The puzzle is a caution against vague conditioning statements; in real problems you must be precise about exactly which event you are conditioning on, because innocent-sounding rephrasings restrict the sample space differently.
Four equally likely families: BB, BG, GB, GG. 'At least one boy' keeps BB, BG, GB; P(both boys) = 1/3. 'The first child is a boy' keeps BB, BG; P(both boys) = 1/2. Same family, different phrasing, different answer.
Specifying which child is a boy changes the conditioned sub-space and the answer.
The clean 1/3 answer assumes the information arose as 'at least one is a boy', not from, say, randomly meeting one child who happened to be a boy — a different procedure gives 1/2. Spell out the sampling.