I currently have a number of problems of the following form:

I have a boolean value random variable Q I want to simulate draws from. I cannot precisely calculate q = P(Q), but I *can* produce a sequence u\_1 <= ... <= u\_n <= ... <= q <= ... <= v\_n  <= ... v_1 with v\_n - u\_n -> 0.

And this is enough to simulate Q:

Draw a random number r between 0 and 1. Run the sequence until we find some n such that r <= u\_n or r >= v\_n. If r <= u\_n return true, else return false.

This works because it ends up equivalent to returning true if r <= q or true otherwise, which is precisely how to simulate Q.