The Two Envelope Paradox and Where the Reasoning Breaks

The two envelope paradox resolves once the hidden, impossible prior behind the always-switch argument is replaced with a real probability distribution.

The two envelope paradox is resolved by recognizing that the argument for always switching secretly assumes a probability distribution on the amount of money that cannot actually exist, an improper prior with no finite normalization. Once you replace that hidden assumption with any genuine, normalizable distribution over the amount of money in the envelopes, the correctly computed expected value of switching turns out to equal exactly the amount already in your hand, matching the plain symmetry intuition that switching should never be worth it before you have opened anything.

You are shown two identical envelopes and told that one contains exactly twice as much money as the other. You pick one envelope and open it, revealing an amount X. You are offered the chance to switch to the other, unopened envelope. The argument for switching runs as follows: the other envelope contains either X divided by 2 or 2 times X, each with probability one half, so the expected value of switching is one half times X divided by 2, plus one half times 2 times X, which works out to 1.25 times X, strictly more than X. By this logic you should always switch, but the identical argument would then apply to the envelope you just switched into, implying you should switch back, and so on…

The step that fails is the claim that, having observed X, the other envelope contains X divided by 2 or 2 times X with equal probability one half, for every possible value of X at once. That claim is only true if the amount in the smaller envelope is drawn from a distribution where every value is equally likely, spread uniformly across all positive numbers with no upper bound. No such distribution exists, since a uniform spread over an unbounded range cannot be normalized to total probability 1. The argument for always switching is built on a prior that is not a real probability distribution, which is why it produces a contradiction.

To resolve the paradox honestly, you need to specify a real, proper prior for how the smaller envelope's amount was chosen, and then apply Bayes' rule to work out the correct probability that your envelope holds the smaller or larger amount, conditional on the specific value X you observed. That conditional probability generally is not one half for both outcomes, and once you use the correct conditional probabilities instead of the naive 50-50 split, the paradox disappears.

Suppose the smaller envelope's amount is drawn from the following proper distribution: it equals $10 with probability one half, $20 with probability one quarter, $40 with probability one eighth, and in general 10 times 2 to the power n dollars with probability one half raised to the power n plus 1, for n equal to 0, 1, 2, and so on. This distribution sums to exactly 1 across all n, unlike the uniform prior the paradox implicitly assumes. Once the smaller amount is drawn, you are equally likely to be handed the envelope with that smaller amount or the envelope with double it.