You are told there are two envelopes. One contains twice as much money as the other one. You pick one but are allowed to change your mind after picking it. (You are equally likely to pick the one with less money as the one with more money.)

To figure out how much on average the other envelope should contain, one might average x/2 and 2x because one is equally likely to pick one as picking the other. Since this comes out to 5x/4, one might always change his or her mind. But wouldn't this end up with one never making up his or her mind?

To me, the answer to this paradox is simply a matter of size.  If the first envelope I pick up and look at has only $1 in it, that means the other has either $2 or 50-cents.  So, if I change my mind, I have a 50% shot at $2 which is better than $1.  If the other envelope turns up with only 50-cents, I will have only lost 50-cents.  On the other hand, if the first envelope has $1000 in it, that means the other envelope has $500 or $2000 in it.  I would not want to lose $500 on the chance of maybe getting $1000 more than I have.  So, I would keep the $1000 envelope. 

Posted by Martin Thompson on 2004-08-18 18:48:43