Which comes first in coin tosses, HHT or HTT

Audrey tosses a fair coin over and over until either the pattern $HHT$ or the pattern $HTT$ appears as three consecutive tosses. What is the probability that $HHT$ shows up first?

$\tfrac23$. Leading tails are irrelevant, so start from a head and look at the next two tosses. $HH$ guarantees $HHT$ will win, because the run of heads must end with $HHT$. $HT$ then $T$ gives $HTT$, and $HT$ then $H$ returns to the starting state. So $HHT$ wins with probability $\tfrac{1/2}{1/2+1/4}=\tfrac23$.