The breakdown for me was in step 4-5. > If A is the smaller amount, then the other envelope contains 2A. > If A is the larger amount, then the other envelope c

The breakdown for me was in step 4-5. > If A is the smaller amount, then the other envelope contains 2A. > If A is the larger amount, then the other envelope contains A/2. These are conditional probabilities so you can't simply add them up and compute an expected value of switching like they show in the example.