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?

Suppose the amount in the lesser envelope is x.  Then the amount in the larger envelope is 2x.  Now, to determine whether you should switch or not, consider the probability of chosing each envelope and the potential gain/loss of then switching.

Case A:  You picked envelope x.  In this case, switching would increase your money by x.

Case B:  You picked envelope 2x.  In this case, switching would decrease your money by x.

Since each case has equal probility, these potential gains/losses are equally weighted: (x)+(-x)=0.  So, we correctly conclude that switching will neither improve or hurt our chances of chosing the larger envelope.

Posted by stan on 2004-05-06 16:59:59