Question

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

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.

Answer 1

yes friend A's argument is incorrect, because before choosing the other envelope the person has to keep his 20 dollar gift card into one envelope and he has to switch for other envelope and if the other envelope has 10 dollar gift card then he loses 10 dollars for switching over other envelope.

the correct calculation is, if the person switches over the other envelope then he will have an average of 15 dollars which is less than the present amount (20 dollars).

so it is better not to switch for other envelope.