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?

The way presented shows what happens when you find the arithmetic mean.  However, if you find the geometric mean (sqrt(x*y)), then it comes out just fine.  Why use the geometric mean instead of the arithmetic mean?  Well, consider this:  If you had a million dollars, would you go to extreme lengths to increase it to a million and one?  If you had only a penny, would you not put more effort into getting that extra dollar?

 

Anyway, nobody (me included) has really disproved the paradox.  Everyone is just showing how to get the correct average.  Well, now that we know that the average presented in the puzzle is wrong, can we find why it is wrong... besides the fact that it gets the wrong answer?

Posted by Tristan on 2004-05-07 22:37:42