Compute win probability in coin-toss game

Q: Compute win probability in coin-toss game

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Question

Two players, A and B, play the following game with a fair coin:

They toss the coin alternately: A tosses first, then B, then A, then B, and so on.
The sequence of results (H for heads, T for tails) is recorded in order.
The game ends as soon as the subsequence "HT" (a head immediately followed by a tail on the next toss) appears for the first time.
The person who tosses the tail in this first "HT" subsequence wins the game.

Assuming the coin is fair and the game continues indefinitely until it ends, what is the probability that player A wins the game?

Compute win probability in coin-toss game

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