Two girls given various clues about the daughters

A family has two children, each independently a boy or a girl with probability $\tfrac12$. Find the probability that both are girls given: (a) at least one is a girl; (b) the older child is a girl; (c) at least one is a girl with a particular very rare name.

(a) $1/3$; (b) $1/2$; (c) just under $1/2$. In (a) only BB is ruled out, leaving GG as one of three cases. In (b) the younger child is independent. In (c), if each girl independently has the name with small probability $r$, the answer is $\tfrac{2-r}{4-r}\to\tfrac12$: the rare name nearly identifies one specific child, much like (b).