The Twin Envelopes

Do not switch — or stay, it does not matter. The expected gain from switching is exactly zero under any valid probability distribution. The apparent $25 gain is a mathematical illusion created by an impossible uniform prior over all positive amounts.

Let the smaller amount in the two envelopes be X, drawn from some unknown distribution. When you see $100, Bayesian reasoning requires knowing how likely it is that X = $100 versus X = $50. These are not automatically equal. Step 1: Suppose the two envelopes hold X and 2X, where X follows some distribution f(x). Step 2: You see $100. The other envelope holds either $50 (if X = $50) or $200 (if X = $100). Step 3: P(other = $200 | you see $100) = P(X = $100) / [P(X = $100) + P(X = $50)]. Step 4: Only if P(X = $100) = P(X = $50) does switching have positive expected value. This equality depends entirely on the specific distribution f(x). Step 5: The paradox assumes this equality holds for every possible observed amount — which requires P(X = k) to be the same for every positive k. That means a uniform distribution over all positive reals. Step 6: No such distribution exists mathematically (it cannot integrate to 1). The paradox is smuggling in an impossible prior. Key Insight: The expected value calculation 0.5×$50 + 0.5×$200 = $125 is only valid if both outcomes are truly equally probable from a real distribution. For any actual valid prior, the expected gain from switching is always exactly zero — the envelope's contents give no useful information without knowing the distribution.