Let's say I have a stochastic process that outputs $1$ or $0$ with probability $p$ or $1-p$ respectively, $p\neq 1/2$. Let's assume this is a repeatable iid process. So I can generate $X\_1,X\_2\dots$ which are each $1$ or $0$ as above. Can I create a proposition (a logical construction out of the $X\_i$s) with probability a half. That is, can I simulate a fair process with this unfair one.

If $p=1$ or $p=0$ then obviously no. But (without loss of generality let's assume $1>p>1/2>1-p>0$. Is there always some such proposition? I can get arbitrarily close, sure. Think about the set of binary strings of length $n$ (you can think of my stochastic process as generating these sequences with certain probabilities). For large enough $n$, even the most likely of them is less than a half, and by picking the right ones to go in my "proposition" I can get as close to $1/2$ as I like, as long as I can increase $n$ as much as I want.

For some values of $p$ you can get to exactly $1/2$. For example, take $p(2-p)=1/2$ and solve for $p$. For this value, $X\_1 \lor X\_2$ has probability exactly $1/2$. Is it always possible to find a proposition that has exactly probability a half? Or is there a characterisation of the values of $p$ that you can do this for?

(Wasn't entirely sure what tags were appropriate)