How about I choose a prisoner at random from among all the prisoners in the problem. What is the probability that the prisoner I have chosen has correctly stated the color of the hat on his head?
So what do you mean to choose a prisoner at random when one has infinitely many prisoners?
Whatever mwengler's answer is, your answer is going to have to be "in that case the set you asked about isn't measurable, and so I can't assign it a probability".
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.