A mathematical problem I can't seem to get Mathematica to auto solve.

[polymath257]

How to map a uniform probability space on an infinite set?

The problem is that the set of natural numbers is countable and so does not have a uniformly distributed measure on it.

That said, the density of a set is often a good alternative. If A is a set, consider the ratio #{x\in A: x<=n} / n as n-->infinity.

While the limit doesn't always exist, this has some of the properties we expect from probabilities.