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I have a series of N binomial (win/loss) events $e_i$ that can happens with different probabilities $p_i$. If i count the times $W$ that an event happens (the number of wins), the mean probability will be $\hat p = \frac{W}{N}$. What is the relation between $\hat p$ and $p_i$?

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In distribution, $\hat p$ can follow a variety of distributions on $\{0,1,\ldots,N\}$ since, for every $0\leqslant k\leqslant N$, $ \mathrm P(N\cdot\hat p=k)=\sum\limits_{|I|=k}\prod_{i\in I}p_i, $ where the sum runs over every subset $I$ of $\{0,1,\ldots,N\}$ of size $k$. In the mean, $N\cdot\mathrm E(\hat p)=\mathrm E(W)=\sum\limits_{i=1}^N\mathrm P(\text{event}\ e_i\ \text{is a win})=\sum\limits_{i=1}^Np_i. $ Finally, by independence, the variance is such that $ N^2\cdot\text{Var}(\hat p)=\sum\limits_{i=1}^Np_i(1-p_i). $

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    *you postulated that the $p_i$ are a sequence of 1 and 0's*... Absolutely not.2012-04-11