Lets say we construct a list as follows. It has f(n) 1's, and f(2) 2's and f(3) number of three's in it etc.
Let L(n,f) be this list, so if f(n)=n^2 we get L(3,n^2)=(1,2,2,2,2,3,3,3,3,3,3,3,3,3)
If f(n) is such that f(m)>(sum f(n) n=1 to n=m-1). Then the probability to pick the number n from the list L(n,f) is above 1/2.
So if we take that f(n), but reverse is such that the smallest number always has the most appearances in the list, we make it into g(f) (we can do this since it was arbitrary), then the probability to pick the nmuber "1" from L(n,g(f)) is above 1/2 for all n, but then we let n->inf, then L(inf,g(f)) contains all the distinc integers, but the probability if you pick a random number from the list that it is 1, is still 1/2, is it possible ?