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 paradox would have been true, had there been a uniform distribution of x from 0 to infinity. in that case, no matter what X would be, there would be exactly 50% for x/2 or for 2x (then the perc. of getting 2x or x/2 would NOT be dependant on X itself). however, no matter what kind of distribution is chosen, there is ALWAYS lower odds for getting 2x than getting x/2, and there is where we are wrong - it's not 50%.
for example, one method of choosing a number from 1 to infinity is randomizing a number from 0 to 1 and taking 1/x. in that distribution e=2, for example, so for each x>2, there's no reason for switching an envelope.
Posted by ronen
on 2004-05-08 06:56:33