Coin toss game II

Two friends named $A$ and $B$ are playing a coin-tossing game. They take turns tossing a fair coin. If the sequence of heads and tails forms a $HT$ subsequence, the game ends and the player who tosses the tail wins. $A$ goes first, followed by $B$, and the game continues in this pattern. What is the chance of $A$ winning the game?