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This answer is a bit late, but the problem has quite an elegant solution based on the Fourier transform and I wanted to add it. The action of the BSC can be modeled as a$\pmod 2$ sum of the data bit, $X$, and a random noise bit, $N$, who's probability distribution is $P(N=0)=1-p$ and $P(N=1)=p$, $$Y = X \oplus N.$$ The distribution over the output, $Y$, ...

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