Discussion question in research interviews.
The two envelopes
The question
Two envelopes contain money, one twice the other. You open one and find 100. Swapping appears to have expected value 125. What is wrong?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
The calculation assumes the other envelope is equally likely to hold 50 or 200 whatever you observe, which no proper prior over amounts allows.
The naive step writes expected value as half of 50 plus half of 200. That requires the conditional distribution given you saw 100 to be an even split.
For that to hold at every possible observation you would need a prior on the smaller amount that is uniform over all scales, which is not a probability distribution.
With any genuine prior, seeing a large amount makes it more likely you already hold the bigger envelope, and the apparent free gain disappears.
Interviewers care less about naming the paradox than about seeing you ask 'what prior makes this calculation legal'.
What it is testing
Whether you notice an unstated and impossible prior.
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