Two envelopes beside a short bounded row of coin stacks. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
A bounded prior makes some alternatives impossible
Let the smaller amount X be 10 or 20 with equal weight. Possible envelope pairs are therefore (10,20) and (20,40). With a uniform initial envelope choice, observed amounts 10 and 40 each have probability one-quarter, while 20 has probability one-half.
Condition separately on each observation
After seeing 10, the other amount must be 20, so the expected change from swapping is +10. After seeing 20, the other amount is 10 or 40 with equal conditional weights, giving expected change +5. After seeing 40, the other amount must be 20, so the change is −20. The upper boundary removes the imagined possibility of 80.
Check the unconditional balance
Weight those conditional changes by their observation probabilities: (1/4)×10+(1/2)×5+(1/4)×(−20) = 0. Thus a positive conditional gain at one observed amount can coexist with zero unconditional gain. There is no contradiction between conditional updating and unopened-envelope symmetry.
Do not invent a prior when the story omits it
The Two Envelopes Paradox Calculator only answers observed-amount questions under the explicitly supplied finite prior. An amount unsupported by that prior is rejected as an impossible observation. Read before versus after observation for the source of the familiar 50:50 mistake. The Bayes' Theorem Calculator provides the general evidence-updating framework, whereas this page works through the two-envelope support and boundary structure.
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