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The Two Envelopes Paradox
The Two Envelopes Paradox is a famous problem in probability theory and decision theory. Two envelopes each contain money, with one containing exactly twice the amount in the other. You pick one envelope and see it contains X dollars. The question is: should you switch to the other envelope?
The apparent paradox arises from this reasoning: the other envelope contains either 2X or X/2, each with probability 1/2. The expected value of switching is (1/2)(2X) + (1/2)(X/2) = 5X/4, which exceeds X. This suggests you should always switch, but then the same argument applies after switching, creating an infinite loop. Something must be wrong with the analysis.
Mathematical Analysis
This calculation is flawed because X is not a fixed value independent of which envelope you hold. If the pair has amounts A and 2A, and you hold the smaller (A), switching gains A. If you hold the larger (2A), switching loses A. The expected gain is zero.
Resolution
The paradox resolves when we properly condition on the underlying pair of amounts. Let the smaller amount be A (unknown). With equal probability you hold A or 2A. The expected value of the other envelope given your value X depends on the prior distribution of A, which the naive calculation ignores. Without additional information about A, switching has zero expected gain.
- If you know the distribution of A, sometimes switching is beneficial and sometimes not
- With no information about A, switching and staying have identical expected outcomes
- The fallacy is treating X as a fixed constant while simultaneously allowing it to be either the smaller or larger amount
Frequently Asked Questions
Should I switch or stay?
Without additional information, it does not matter. The expected gain from switching is zero when properly analyzed. The paradox lies in the flawed calculation that treats the observed amount as fixed while simultaneously acknowledging it could be either the smaller or larger value in the pair.
Is this related to the Monty Hall problem?
Both are famous probability paradoxes, but they are fundamentally different. In Monty Hall, new information is revealed (the host opens a door), which makes switching beneficial. In the Two Envelopes problem, no new information is gained, so switching provides no advantage.