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.