Coin Toss Game (HT)

Two players, $A$ and $B$, alternatively toss a fair coin ($A$ tosses the coin first, then $B$ tosses the coin, then $A$, then $B$…). The sequence of heads and tails is recorded. If there is a head followed by a tail ($HT$ subsequence), the game ends and the person who tosses the tail wins.

Question: What is the probability that $A$ wins the game?