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Now suppose the person reasons as follows:
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| type = decimal|Denote by ''A'' the amount in the player's selected envelope.|The probability that ''A'' is the smaller amount is 1/2, and that it is the larger amount is also 1/2.|The other envelope may contain either 2''A'' or ''A''/2.|If ''A'' is the smaller amount, then the other envelope contains 2''A''.|If ''A'' is the larger amount, then the other envelope contains ''A''/2.|Thus the other envelope contains 2''A'' with probability 1/2 and ''A''/2 with probability 1/2.|So the [[expected value]] of the money in the other envelope is
<math display="block">{1 \over 2} (2A) + {1 \over 2} \left({A \over 2}\right) = {5 \over 4}A</math>|This is greater than ''A'' so, on average, the person reasons that they stand to gain by swapping.|After the switch, denote that content by ''B'' and reason in exactly the same manner as above.|The person concludes that the most rational thing to do is to swap back again.|The person will thus end up swapping envelopes indefinitely.|As it is more rational to just open an envelope than to swap indefinitely, the player arrives at a contradiction.
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