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Why is it necessary to say that the two envelopes come from a distribution? Essentially, we can approach the two envelope problem in one of two ways. Firstly, we can treat it as a fantasy in which anything can happen without any rational rules, ie the two envelopes appear magically out of nowhere, with no probabilities attached, and hence there is no possibility of making a rational calculation. This is like asking what is the chance that Lance Armstrong is thinking about grated carrots at this moment. Such a question cannot be answered mathematically to produce an answer in which we could have confidence. Put simply, if there is no framework within which a probability calculation can be made then we cannot make it.

The second choice is to treat the puzzle as a situation that can be analysed using probability theory. If we are to make a probability calculation then we need to make a basic assumption - that our sample (the two envelopes) comes from some kind of distribution, ie it comes from a well-defined range of probabilities. We do not know the nature of this distribution, and we should not make any assumptions about it, but we have to assume that it exists in order to calculate a result for the expected gain of switching envelopes.

What is meant by a distribution? Here are two examples, the first finite, the second infinite:

(I) ( 100, 200 ) with probability 37% and ( 6, 12 ) with probability 63%.

(II) (1, 2), (2, 4), (4, 8), (8, 16), (16, 32), ... with the probability of the pair ( 2n, 2n + 1 ) being 2-n -1 for all integers n >= 0. Note that the probabilities in every possible distribution must add up to 1.

The general case that we must address while solving the two envelope paradox is the collection of all possible distributions like the two given above. We need to calculate a double sum, firstly over all possible distributions of two envelopes, and secondly over all possible choices of two envelopes within each of those distributions.

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