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「Terence Tao "one’s intuition on probability should not be trusted here”」で検索
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http://math.stackexchange.com/questions/886180/formal-approach-to-countable-prisoners-and-hats-problem
probability theory - Formal approach to (countable) prisoners and hats problem. - Mathematics Stack Exchange: asked 2 years ago asked Aug 3 '14 at 10:25
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I've found this nice puzzle about AC (I'm referring to the countable infinite case, with two colors). The puzzle has been discussed before on math.SE, but I can't find any description of what is happening from a formal point of view.
I'm not really into probability theory, therefore I apologize in advance if I do any mistake or if I can't understand something which is obvious. In particular, I don't know much about infinite sequences of random variables.

Intuitively, the solution is quite paradoxical, and this seems to be the reason: it seems that each prisoner has 50% chance to go free and 50% chance to be killed and nothing (i.e. no strategy) can change this probability,
since each prisoner gets no data about his hat from the others and from "the environment". Furthermore, every prisoner's guess is independent from the others. Thus, for the way we intuitively think about probability, it seems that the expected value of prisoners going free should be "a half of N
" (whatever this means). It turns out that (using AC) there exists a strategy which allows all but a finite number of prisoners go free (and for sure this is not "a half of N", whatever this means).
つづく