A family has two children and each child is equally likely to be a boy or a girl. Given that at least one of the children is a girl, what is the probability that both are girls?

The equally likely families are BB, BG, GB, GG. Knowing at least one girl removes BB and leaves 3 cases, of which only GG has two girls. So the probability is 1/3, not 1/2 — the information about one child changes the sample space.