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Suppose I have three biased coins, which I call $C_1$, $C_2$ and $C_3$. They each have probability of heads $p_1$, $p_2$, and $p_3$ (also assume that each flip is independent of another). I use them to run the following process:

For $n$ iterations, where $n$ is very large:

1) Flip coin $C_1$ with probability of heads $p_1$.

2) If the result of the flip of $C_1$ in the first step is heads, then I flip $C_2$ with probability of heads $p_2$ and record the result (heads or tails).

3) If the result of the flip of $C_1$ in the first step is tails, then I flip $C_3$ with probability of heads $p_3$ and record the result (heads or tails).

I am interested in the distribution of the number of heads $X$ (out of $n$ trials). This seems like a relatively simple problem and I can write a Monte-Carlo experiment for it, however, my $n$ is very large (and I have lots of instances of $p$'s) so I am wondering if there is an easily computable analytic expression for an approximation (or exact result) for the distribution of $X$...

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The probability to get heads in any given iteration is $p=p_1p_2+(1-p_1)p_3$. The results of the iterations are independent. Hence the number of heads in $n$ iterations is binomial $(n,p)$.

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