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Possible Duplicate:
If a player is 50% as good as I am at a game, how many games will it be before she finally wins one game?

Can anyone help me solve the following problem:

Player A and Player B are playing a game with multiple rounds. The game stops once one of them wins $10$ rounds and is declared the winner. Player A's chances of winning in each round are $\frac13$. What are Player A's chances of winning the game?

Thank you!

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    To be precise, the question isn't an exact duplicate, but some of the answers are.2012-08-09

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We assume independence, which for certain sports may not be realistic. Modify the game by stipulating that whether or not somebody wins $10$ rounds earlier, the game goes on to $19$ rounds. Then A wins the original game if and only if she wins $10$ or more rounds in the modified game. Finding the probability of this is a straightforward "binomial" problem. The answer is $\sum_{k=10}^{19}\binom{19}{k}\left(\frac{1}{3}\right)^k\left(\frac{2}{3}\right)^{19-k}.$ Getting a numerical answer out of this by hand is a little unpleasant. Some calculators and many programs can handle it easily.

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    Thanks! I just got a bit confused because in the question you linked above, the computed value was 0.9352 but the chances of the girl winning the game was 7%. If I understand right, her odds were 7% based on the statement "she should win about one game out of... 15 games." In hind sight, I realized that 93% was the guy's odds. Haha2012-08-09