Should you switch?
I prepare an envelope containing a certain amount of money, and hand it to you. You open it and count how much money is inside.
I then go in another room and prepare a second, sealed, envelope, containing either half the amount of money as in the envelope you hold or twice the amount of money as in the envelope you hold, the chances to be determined by the toss of a fair coin. You do not see the toss of the coin or how much money I have put in the second envelope.
I then come back into the room where you are patiently waiting, and give you a chance to swap the envelopes. If you do, you must take the amount of money in the sealed envelope.
Should you switch envelopes?
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1 solution
The amount of money you hold is 2N. If you accept the second envelope, you have a 50% chance of receiving N and a 50% chance of receiving 4N. So your mathematical expectation is to receive 2.5N. So, you should switch.