Two closed ivory envelopes beside one partly opened envelope with blank tokens. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Before opening, symmetry gives equal expectations
One envelope contains X and the other 2X. Choose an envelope uniformly before observing either amount. Conditional on X, the expected chosen amount is (X+2X)/2 = 1.5X, and the other envelope has the same expectation. With a finite prior over X, both unopened expectations are 1.5E[X].
An observed amount creates a conditioning question
After seeing y, the alternative amount could be 2y if your envelope was the smaller one, or y/2 if it was the larger one. Those possibilities do not automatically have equal posterior probability. Their weights depend on how likely the underlying smaller amounts X=y and X=y/2 were under the model.
Check an explicit two-value prior
Suppose X is 10 or 20 with equal prior weight. Observing 20 can come from the larger envelope in the X=10 pair or the smaller envelope in the X=20 pair. These pathways have equal weight, so the other envelope has conditional expectation (10+40)/2 = 25. This conditional result does not imply a universal benefit from switching before an amount is seen.
Keep the prior visible
The Two Envelopes Paradox Calculator requires a finite prior and a uniform initial envelope choice. Read finite-prior boundary effects to see why other observed amounts behave differently. The existing Expected Value Calculator handles a supplied payoff distribution; this tool focuses specifically on the missing conditional weights behind the paradox.
Check the model behind your result
Use the Two Envelopes Paradox Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26