>>116

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The only comment which does really make sense to me is Terence Tao's one:
This paradox is actually very similar to Banach-Tarski, but involves a violation of additivity of probability rather than additivity of volume.

Consider the case of a finite number N of prisoners, with each hat being assigned independently at random. Your intuition in this case is correct: each prisoner has only a 50% chance of going free.
If we sum this probability over all the prisoners and use Fubini’s theorem, we conclude that the expected number of prisoners that go free is N/2. So we cannot pull off a trick of the sort described above.

If we have an infinite number of prisoners, with the hats assigned randomly (thus, we are working on the Bernoulli space ZN2), and one uses the strategy coming from the axiom of choice,
then the event Ej that the j^th prisoner does not go free is not measurable, but formally has probability 1/2 in the sense that Ej and its translate Ej+ej partition ZN2 where ej is the j^th basis element, or in more prosaic language,
if the j^th prisoner’s hat gets switched, this flips whether the prisoner gets to go free or not. The “paradox” is the fact that while the Ej all seem to have probability 1/2, each element of the event space lies in only finitely many of the Ej.

This can be seen to violate Fubini’s theorem ? if the Ej are all measurable. Of course, the Ej are not measurable, and so one’s intuition on probability should not be trusted here.
(Terence Tao's 終わり)
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