Which comes first, HTH or HHT

A fair coin is tossed repeatedly and the results are recorded. You win if the pattern HTH shows up before the pattern HHT; if HHT shows up first, your friend wins. Find the probability that your friend is the winner.

$\tfrac23$. Both patterns pass through HT. Look at the first HT: if the toss before it was H, HHT has already appeared (probability $\tfrac12$); otherwise HTH needs the next toss to be H (probability $\tfrac14$), and on a T the race effectively restarts. So HHT wins with probability $\tfrac{1/2}{3/4}=\tfrac23$.