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I randomly pick one of two coins (with equal weight) from a bag. Flipping coin 1 has a probability of $p_1$ of yielding heads. Flipping coin 2 has a probability of $p_2$ of yielding heads. Let $I$ be the indicator variable for whether coin 1 was picked (so $E(I) = \frac{1}{2}$). I flip the picked coin $n$ times. Let $X$ be the total number of heads out of $n$ that result.

What is $\mathrm{Var}(E(X\mid I))$?

Calculating $E(X\mid I)$ is not too bad I think: $E(X\mid I)$ is $np_1$ since $E(X\mid I)$ is distributed $\mathrm{Binomial}(n, p_1)$. However, how do I calculate the variance of $E(X\mid I)$ (not of $X\mid I$, but of $E(X\mid I)$)?

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One thing that confused me when I started learning probability should be stated: While the expression $E(X)$ stands for a "final" value, the expression $E(X|I)$ is again a random variable. – Yoni Rozenshein Dec 12 '12 at 9:07

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