You have won a prize. There are two envelopes that each contain a gift card. One gift card amount is exactly double the other gift card amount. You are given one of the two envelopes based on the result of a fair coin flip. You open it and find a $20 gift card.

Now, you are given a choice. You can either keep the $20 gift card, or you can switch for the other envelope. If you switch, you will keep the gift card in the other envelope. Before you decide, a friend A gives you some input:

"Your envelope has a 50-50 chance of containing the smaller or larger gift card amount. If your envelope contains the smaller amount, then the other has $40. If your envelope contains the larger amount, then the other has $10. So the expected value of the amount you will have by switching envelopes is

1210+1240=25 By switching envelopes, you will have on average $25 which is more than you have now, so you should switch!"

Friend B is skeptical:

"Friend A's argument implies that the other envelope is always better, on average. Yet, you received your envelope as the result of a fair coin flip. How could the other envelope always be better on average?"

Explain why Friend A's argument is incorrect, and do the correct calculation.

Hint: Start out with a sample space. The numbers 20, 10, and 40 should not appear anywhere in the sample space, since initially none of this is known. Instead, you could use amounts x and 2x. The probability your envelope contains x is 1/2, and the probability your envelope contains 2x is 1/2. Try the calculation now.

Friend A claims the other envelope has a 50% chance to be $40 and a 50% chance to be $10. Why is this not possible in the sample space that we have?