Consider, to begin with, a simple sort of Two Envelopes scenario: Mr. E. writes out a cheque for £18, another for £12, and puts each inside an envelope. Showing the two envelopes to his friend, Miss Take, he tells her that they contain money, and asks her to take one as a gift. Miss Take does so, and finds that she has £12. Then Mr. E. tells her that one of the cheques was for 50% more than the other, and asks her if she wants to swap her cheque for the other one. Now, Miss Take knows that either the amount in the other envelope is £12 + £6 = £18, or else it is £8 (since £8 + £4 = £12). So she knows that by swapping she would either gain £6 or lose £4. And she also knows that by swapping she is as likely to gain as to lose, since she has no idea which envelope contained the larger amount. So it seems to her that if she swaps she is 50% likely to gain £6 and 50% likely to lose £4. In other words, the mathematical expectation from swapping is 50% of £6 minus 50% of £4, which is £1. Since that is a positive amount, Miss Take decides to swap.
......She feels justified when she ends up with £18 but of course, she had no real reason to swap because her original choice was blind, and by swapping she is in effect just making that choice again. Still, when she swapped she did have more information about what was in the envelopes, and she seems to many to have had some mathematical reason to swap. Such is the Two Envelopes problem, which has been much discussed following M. Kraitchik’s two wallets (in 1953; for a translation of the relevant passage, see previous link). The discussion usually revolves around the problem of randomly choosing the amounts in the envelopes (with one being twice the other, usually), but it clearly does not matter how Mr. E. came by the amounts of £18 and £12. Maybe it was the 18th of December, and Miss Take’s birthday; or maybe Mr. E. was a fan of Tchaikovsky’s 1812. The main thing was that Miss Take did not know both amounts, and that the envelopes did not reveal which contained the larger amount.
......To give us some perspective on Miss Take’s mistake, Mr. E. then offers an identical pair of envelopes to his colleague, Miss Tree, who also picks the one containing £12. But this time, when he offers her the chance to swap envelopes, he tells her that the square of one third of the larger amount is twelve plus twice the smaller amount (in £s). Miss Tree calculates that either the amount in the other envelope is £18 (since 6 is one third of 18, and 6 squared is 12 plus twice 12), or else it is £2 (since 4 is one third of 12, and 4 squared is 12 plus twice 2). So swapping would either gain her £6, or else it would lose her £10. And for all she knows, a gain is as likely as a loss. So now the mathematical expectation appears to be of a loss, of £2. Even so Miss Tree swaps envelopes, precisely because a gain is as likely as a loss. Having read P. N. Johnson-Laird, Miss Tree knows how easily we can be misled by disjunctions; and she suspects that, since only one disjunct of any true disjunction—such as either a gain of £6 or else a loss of £10—needs to be true, so Mr. E. could, by telling her something true about how the two amounts are related, have led her to calculate practically any mathematical expectation.
......The problem of how we should apply mathematical expectations is neither unimportant nor uninteresting, a clearer way to consider randomly choosing a natural number may therefore be via the following variant of a paradox attributed to P. Levy (by F. P. Cantelli, in 1935), which also resembles Kraitchik’s puzzle: Suppose that a god selects two natural numbers at random—i.e. any of them might be chosen, with each being neither more nor less likely to be chosen than any other—and then writes two notes promising the bearer upon demand that many days in paradise, puts them into two identical envelopes, and asks you to pick an envelope. The puzzle is that, whichever you pick, the other is almost certain to contain a far greater gift. That is because, given any natural number, there are only finitely many natural numbers that are smaller than it, and infinitely many that are bigger. But of course, each envelope is as likely as the other to contain the larger amount. Such a random choice therefore seems to be impossible.
......Paradoxically there are, quite plausibly, random real numbers, if the standard set-theoretic axiom of infinity is not too unrealistic; and that is paradoxical because if a god could choose real numbers at random then, since any countable list of real numbers amounts to a correlation between some real numbers and the natural numbers, so a god who knew several such lists might decide that if two randomly chosen real numbers were on the same list then he would present you with the corresponding two envelopes. Regarding the aforementioned plausibility, note that any particle that could decay is, if considered one half-life into the future, mathematically akin to a perfectly fair coin-toss, whence an endless sequence of such particular half-lives could instantiate a random real number in binary notation (with decays corresponding to 1s and non-decays to 0s); and such sequences plausibly exist if space is infinite, or if it lasts forever, or if there are other universes alongside ours (in a multiverse), and so forth...
(PS: This post is linked to in the Philosophers' Carnival 105:)