Alternating tosses: first to complete heads then tails

Two players, A then B, take turns tossing one shared fair coin, so the tosses form a single sequence. The game ends the first time a tail immediately follows a head in that sequence, and the player who tossed that tail wins. What is the probability that A wins?

$4/9$. Track whose turn it is and whether the last toss was a head. If A is about to toss after a head, A wins with probability $b$ where $b=\tfrac12+\tfrac12\cdot\tfrac b2$, so $b=\tfrac23$. From the start, $a=\tfrac12\cdot\tfrac b2+\tfrac12\left(\tfrac b2+\tfrac a2\right)$, which gives $a=\tfrac49$.