The envelope and the banknotes

1. The failure is not in adding hopes, but in how the probabilities are being assigned.
2. Seeing €20 is not enough, by itself, to treat the two remaining possibilities as symmetrical.
3. Before deciding whether to change, distinguish between the observed quantity and the way the envelopes were generated.

Answer: It cannot be concluded, just from seeing 20 €, that changing is better. Explanation: The temptation is to reason like this: in the other envelope there is either €40 or €10, each case with probability \(1/2\). Then its expected value would be \[
\frac12\cdot 40+\frac12\cdot 10=25,
\] and it would seem that it is appropriate to change. The problem is not in the account, but in the previous assumption. After opening an envelope and seeing €20, you still don't know if that envelope was the big one or the small one, but those two possibilities don't have to be equiprobable. To decide, it would be necessary to know how the initial quantities were chosen or what probability distribution governs the contents of the envelopes. Without this prior model, the observation “I saw €20” is not enough to deduce that the other envelope has a higher expected value. Therefore, with the information given, there is no solid mathematical reason to prefer to change. The so-called “paradox” arises from assigning probabilities of \(1/2\) without justification to two cases that are not defined in that way.