Coin Toss Game | Explaining Greenbook Probability Problems

I am explaining every Quant Greenbook probability problem so I can practice for interviews. Coin Toss Game Problem: 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. What is the probability that A wins the game?