But if you are one of the little people perched atop a cube, and you know these two facts, there is still a third piece of information you need to make predictions: “Which cube am I standing on?”
This nicely illustrates why discrete uniform probability distributions (as I understand them) over infinite sets don’t work very well.
I can’t make sense of this thought experiment. I’ll dump my reasoning below, and would be grateful for any clarifications about what I’m doing wrong.
Assume I’m one of the people on one of those cubes, know about the entire series including the people on them, and haven’t looked at the number below me yet.
What’s the probability I’m on the first cube, 1? Well, that’s one possibility, and there’s … countably infinitely many … alternatives, so if that probability isn’t zero, it’s as close as I can make it. The same reasoning applies to every other cube.
I know all of the cubes exist, each has a person, and I’m one such person. If this is all I know, I have no particular reason to assign a non-uniform probability distribution over the possible outcomes. So, since I will assign the same probability to finding myself on each of the cubes, that leaves me with the following options:
1) I can assign a probability of zero, which blows up in my face since I have to conclude I won’t find myself on any of the cubes.
2) I can assign a non-zero probability, which blows up in my face since by summing those probabilities I will necessarily get a total probability of greater than one (or any finite number, for that matter).
But if you are one of the little people perched atop a cube, and you know these two facts, there is still a third piece of information you need to make predictions: “Which cube am I standing on?”
This nicely illustrates why discrete uniform probability distributions (as I understand them) over infinite sets don’t work very well. I can’t make sense of this thought experiment. I’ll dump my reasoning below, and would be grateful for any clarifications about what I’m doing wrong.
Assume I’m one of the people on one of those cubes, know about the entire series including the people on them, and haven’t looked at the number below me yet. What’s the probability I’m on the first cube, 1? Well, that’s one possibility, and there’s … countably infinitely many … alternatives, so if that probability isn’t zero, it’s as close as I can make it. The same reasoning applies to every other cube. I know all of the cubes exist, each has a person, and I’m one such person. If this is all I know, I have no particular reason to assign a non-uniform probability distribution over the possible outcomes. So, since I will assign the same probability to finding myself on each of the cubes, that leaves me with the following options:
1) I can assign a probability of zero, which blows up in my face since I have to conclude I won’t find myself on any of the cubes. 2) I can assign a non-zero probability, which blows up in my face since by summing those probabilities I will necessarily get a total probability of greater than one (or any finite number, for that matter).