Gambler's Ruin — Stopping Time

Two players, A and B, compete in a coin-flipping game.

• Player A starts with $a$ tokens,
• Player B starts with $b$ tokens.

At each turn, a fair coin is flipped:

• If A wins the flip, they take one token from B,
• Otherwise, they lose one token to B.

The game continues until one of the two players holds all the tokens.

What is the probability that player A wins the game?