Maybe this way then. I set down somewhere in the universe in a location that I don't reveal to you ahead of time and then identify the 100 prisoners that are closest to my position. If the 100th and 101st furthest prisoners from me are exactly equally distant form me I set down somewhere else in the universe and I keep moving until I find a location where I can identify the 100 closest prisoners to my current position.

Of that 100 prisoners, I count the number of prisoners who identified their hat color correctly.

My question is what is the probability that I have counted 50 or fewer correct answers? Is it greater than, less than, or equal to the probability that I have counted 51 or more correct answers?

Thanks to you and JoshuaZ for trying to help me here.

Is it safe to say that this problem, this result, has no applicability to any similar problem involving a merely finite amount of prisoners, say a mere googol of them?

Yes. But I do think that thinking critically about the assumptions you are making, in particular that you can meaningfully talk about what it means to pick a random individual in a uniform fashion, is worthwhile for understanding a fair bit of probability and related issues which are relevant in broader in contexts.