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Aa n dB play a series of games which ...

`Aa n dB` play a series of games which cannot be drawn and `p , q` are their respective chance of winning a single game. What is the chance that `A` wins `m` games before `B` wins `n` games?

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For this to happen, A must win at least m out of the first m + n - 1 games. Therefore, the required probability is `""^(m+n-1)C_(m)p^(m)q^(n-1)+""^(m+n-1)C_(m+1)p^(m+1)q^(n-2)+...+""^(m+n-1)C_(m+n-1)p^(m+n-1)`
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