This question is a bit less than rigorous, but it's only because I don't know how to formulate it rigorously.
Suppose there was some machine, or function, or whatever that could output a random positive whole number. Let's say that it has done its job and I have a number n.
Here's what seems like a paradox to me: if n really was randomly generated, I shouldn't be surprised by its value, no matter how big or small. After all, there is no number that I should have expected more than any other.
On the other hand, if I asked the question of whether I expected the number to be bigger or smaller than the one that I got, the answer would always be bigger. In fact, it was infinitely more likely that I would have gotten a number bigger than what I got. If I got a digit with one billion digits, there are still infinitely more numbers above n than there are below it. No matter the number, it is infinitely improbable that it could have been that small.
Is it just that the idea of picking randomly from an infinite set doesn't make any sense? Or is this a problem that one also runs in to when trying to pick a random element of a finite set?