How this calculator works
Before opening, a uniformly chosen envelope has the same expectation as the other envelope. After observing x, the possible smaller amounts are x or x/2, and their prior weights determine the conditional probabilities. The calculator assumes a uniformly chosen envelope for each pair and an explicit finite prior, then renormalizes the cases consistent with the observed value. It does not assign a blanket one-half probability to the other envelope containing x/2 or 2x. Monetary matches allow relative arithmetic drift of 1e-12.
Formula
Unopened pair (s,2s): E(keep) = E(switch) = 3s/2. Observed x: weight prior(s=x)/2 on other=2x and prior(s=x/2)/2 on other=x/2; normalize the surviving weights to obtain E(switch | x).
Worked example
If smaller amounts 5 and 10 have equal prior weight and you observe 10, the pair could be (5,10) or (10,20). The other amount is then equally likely to be 5 or 20, with expectation 12.5 and switching gain 2.5. Observing 20 instead identifies the pair (10,20), so switching gives 10.
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Frequently asked questions
Why is the usual 1.25x argument incomplete?
It assumes equal conditional chances for the other amount being half or double without specifying a prior. Seeing an amount can change those chances.
Can switching help after opening?
Under a stated prior, yes for some observations. Other observations can favour keeping, preserving the equal unconditional expectations in a finite model.
Does this settle every version of the paradox?
This tool handles a finite prior and finite expectations. Infinite-prior versions require additional care about existence of expectations and are outside this model.