Alice's toss count when Bob needs more tosses

Alice and Bob each toss their own fair coin until it first shows heads. Given that Bob needed strictly more tosses than Alice, what is the expected number of tosses Alice made?

$4/3$. $P(A=n,\,B>n)=2^{-n}\cdot 2^{-n}=4^{-n}$, which sums to $\tfrac13$. So $A$ given $A<B$ is geometric with ratio $\tfrac14$: $E=\tfrac{\sum n4^{-n}}{1/3}=3\cdot\tfrac{4}{9}=\tfrac43$.