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I would like to know a formula for the probability of winning a certain number of games.

What is the probability of me winning 2 games if I have a 33.333333333333...% chance of winning each time and I play 5 times?

What is the probability of me winning 2 or more games if I have a 33.33333333333...% chance of winning each time and I play 5 times?

Those are specific cases. I would like to know if there is a general formula for both.

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why don't you just say you have a $1/3$ chance of winning each time? –  Stefan Smith Nov 28 '13 at 3:31

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Suppose the probability of winning a particular game is $p$. You can win $m$ games out of $n$ games you played in $\binom{n}{m}$ ways, and you have to lose the remaining $n-m$ games. The probability of losing a particular game is obviously $1-p$. Then the probability that you win $m$ games out of $n$ games is $$\binom{n}{m}p^m(1-p)^{n-m}$$ The probability that you win $m$ or more games among $n$ games is then given by $$\sum_{t\geq m}\binom{n}{t}p^t(1-p)^{n-t}$$ Now put $n=5,m=2,p=1/3$, to get your answers.

For further reference see the book http://www.amazon.com/Introduction-Probability-Theory-Applications-Edition/dp/0471257087 and look for http://en.wikipedia.org/wiki/Binomial_distribution

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