Consecutive Wins Game
Alice and Bob play a points-based game. Suppose $0 < p < 1$. The game consists of up to $3$ points:
- On the first point, Alice wins the point with probability $p$ and Bob wins with probability
- p$.
- On each subsequent point, the winner of the previous point wins again with probability $p$, and the loser wins with probability
- p$.
A player wins the game as soon as they win two consecutive points. If all $3$ points are played and no player has won two in a row, the entire game is replayed from scratch under the same rules, repeating until someone wins.
Find the probability that Bob wins the overall game when $p = 1/3$.
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