First to have tossed both faces wins with alternating turns

Alice and Bob share a fair coin and take turns tossing it, Alice first. The first player whose own tosses include at least one head and at least one tail wins. What is the probability that Alice wins?

$2/3$. Neither can win on their first toss. From then on, on each turn the tosser wins with probability $\tfrac12$. Alice tosses first in that phase, so $P=\tfrac12+\tfrac14P$, giving $P=\tfrac23$.