Two Envelopes Paradox Calculator

The two envelopes paradox is a famous puzzle in probability and decision theory. You hold one of two envelopes, each containing money, where one envelope has twice the amount of the other. Naively calculating the expected value of switching seems to always favor swapping — yet the same logic applied to the other envelope also favors swapping, which is a logical contradiction. The paradox reveals subtle errors in applying expectation formulas without a proper probability model.

If your current envelope holds amount X, and you assign probability p to the other being double and probability q to it being half, the expected value of switching is: E(switch) = p × 2X + q × X/2. With p = q = 0.5, this gives 1.25X — which is greater than X, naively suggesting you should always switch. However, this reasoning contains a flaw: you cannot simultaneously treat X as both a fixed value and a random variable drawn from the same prior.

The fallacy lies in treating the amount in your envelope as a fixed known value X while simultaneously applying a symmetric probability argument that only holds when X itself is a random variable. In a proper Bayesian treatment, once you observe X, the probabilities of the other envelope being 2X versus X/2 are not generally equal — they depend on your prior distribution over envelope amounts. So the naive 50/50 split is usually not justified.

Not necessarily. The naive calculation suggests always switching, but this leads to an infinite regress — if you switched, you'd want to switch back for the same reason. The correct resolution is that with a proper prior distribution on envelope amounts, the expected gain from switching averages out to zero. Whether to switch in practice depends on your actual probability beliefs about the envelope amounts, which is exactly what this calculator lets you explore.

This calculator requires that the probability the other envelope is double plus the probability it is half should sum to 100% for the model to be consistent. If they don't sum to 100%, the remaining probability is unaccounted for and the expected value calculation may not reflect a valid probability distribution. The calculator will flag this and still compute results, but interpret them with caution.

The expected gain or loss is simply the difference between the expected value of switching and the amount currently in your envelope. A positive value means switching is mathematically favorable under your probability assumptions; a negative value means keeping your envelope is better. A value near zero means it makes no difference either way.

If you believe the other envelope is more likely to be double (e.g., 70% chance), the expected value of switching rises significantly and switching becomes clearly favorable. Conversely, if you think it's more likely to be half (e.g., 70% chance), keeping your envelope is better. This calculator lets you dial in any probability split to see exactly how sensitive the recommendation is to your beliefs.

A random variable is a quantity whose value is determined by a random process. In the two envelopes problem, the amount of money in the unchosen envelope is a random variable — its value is uncertain from your perspective. The paradox partly arises from confusing the observed fixed value of your envelope with a random variable that can be simultaneously X/2 and 2X relative to the other envelope.