The Two‑Envelopes Paradox presents a seemingly simple decision problem: you are given two sealed envelopes, one containing twice the amount of money as the other. After randomly selecting one envelope and observing its amount (A), you must decide whether to keep it or switch to the other envelope.
A naïve expected‑value calculation suggests that switching is always advantageous. Assuming a 50 % chance that the unseen envelope holds the larger amount (2A) and a 50 % chance it holds the smaller amount (frac{A}{2}), the expected value of switching becomes:
However, this reasoning overlooks the fact that the observed amount (A) already encodes information about which envelope was originally chosen. Properly accounting for the prior distribution of envelope values resolves the paradox, showing that the expected gain from switching is actually zero when the amounts are drawn from a realistic distribution.
What is the Two Envelopes Paradox?
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