Puzzler -- Envelope please
Alex Y
This "puzzle" is how to explain the apparent paradox of this situation.
You are shown a table with two index cards labeled "Step 1" and "Step 2" and two unlabeled envelopes.
You turn over card number 1 and read that you may pick whichever envelope you want and keep its contents, then read card 2.
You pick an envelope by coin toss and open the envelope, which contains $100.
Now you read card 2, which says that one of the envelopes originally contained exactly twice as much money as the other, and you may switch the money from the envelope you chose for the other envelope, if you so desire.
You reason as follows: There is a 50/50 chance that I picked the larger envelope to begin with . It costs me $100 to swap, and there is a 50% chance I will end up with $200 and a 50% chance I will end up with $50.
.5*$200+.5*$50= $125, so I should make the switch.
Your gut probably tells you this is wrong, and your gut is right -- in fact, with this logic, you don't even need to open the first envelope, just swap it once you have picked it. But then that is the same as picking the second one, so swap it back, and get rich swapping envelopes ad infinitum!
But what is wrong with the expected return reasoning that calls for a switch?