Choose a goal probability B in the range (0,1). It can't be at 0 or 1, but can be arbitrarily close.
Now let's start sampling, and we'll take this all the way to the limit. For any discrete sample size as sample size goes to infinity, let's choose some positive/negative outcomes such that our probability approaches B. Remember that because our goal probability is greater than 0 and less than 1, there are infinite positive/negative outcomes to choose from.
This distribution has some interesting properties. The derived probability is B. The sample will grow to include all points. There are infinitely many functions to generate this outcome.