A family has two children. You learn that at least one is a boy. What's the probability both are boys?
Most say 1/2, but it's 1/3! The possible families with at least one boy are: BB, BG, GB. Only 1 of 3 is BB.
But if you learn the older child is a boy, then: BB, BG. Now it IS 1/2! The extra information about which child changes everything.