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Suppose one has a biased coin $C_1$ with probability $p$ of landing heads and $(1-p)$ prob. of landing tails.

If one wants to sample a coin $C_2$ with $p^2$ probability, one can flip the coin $C_1$ twice and logically AND the results (assign heads to 11 $C_1$ flips only, otherwise tails). Hence the logical AND corresponds to multiplication of the sampled probability distributions ($C_{1,1} \wedge C_{1,2}$ is sampled as $p \times p = p^2$)

What algebraic operation does the logical OR of the results correspond to in this case? What about other logical operations ($\implies$ and the reverse in particular)?

EDIT: Although I already accepted the answer below, the main motivation for this (very unfortunately laid out) question was meant to be to find out a way of sampling $p + p$ or more importantly $f(p) + g(p)$ for some bias functions $f, g$. The OR composition for example gives $f(p) + g(p) - g(p)f(p)$ - I was thinking if I could eventually come up with a way to approximate the sum operation by relating the boolean operations on the output to algebra.

Thanks, Vojta

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The logical OR would in this case have $0$ only if both tosses land on a zero, so $P(0) = (1-p)^2$ and $P(1) = 1-(1-p)^2$.

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  • $\begingroup$ I got quite lost in oversimplifying the question - but no matter that, your answer is absolutely correct. Thanks! $\endgroup$ – vojta havlíček Jul 29 '14 at 10:39

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