i think you don't understand the question... its not asking what percentage of the time you gain money, and what percentage of the time you lose money. that is 1/2. we see 100$, and there is a 1/2 chance we picked the envolope with the larger amount, and a 1/2 chance we picked the envolope with the smaller amout. So if we switch, half the time we will get 50$, and half the time we will get 200$. 50$ + 200$ = 250$ / 2 = 125$, so a 25$ profit on average from switching the problem is that the above can't be true. That is saying that if we pick envolope A, look at the amount of money in it and switch to envolope B, is more profitable then just picking envolope B the first time, and not switching...