Race to three heads or three tails

A fair coin is tossed repeatedly. Player one wins as soon as three heads have appeared in total, player two as soon as three tails have appeared in total; the heads or tails need not be consecutive. What is the expected number of tosses before someone wins?

$\tfrac{33}{8}$. The game lasts 3, 4 or 5 tosses. It ends at toss 3 with probability $\tfrac14$, at toss 4 with probability $2\cdot\tfrac{3}{16}=\tfrac38$, and otherwise at toss 5 ($\tfrac38$). The mean is $\tfrac34+\tfrac{12}{8}+\tfrac{15}{8}=\tfrac{33}{8}$.