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?
"One might average X/2 and 2X." One might not do this and average correctly. I mean, I understand the reasoning that these are the values relative to the observer. However, it is the way the variable X is used and defined that is ambiguous. It appears with this averaging policy, there is an amount possibly 4 times as much! There is no such amount. There is only such a probablity! The amounts were fixed when the envelopes were sealed.
There are 2 possible outcomes in the begginning. X and X/2 or X and 2X depending on how you define your X. These values do not change and nor does your variable X. Once the envelope has been chosen, the variable X has changed.
Say 1$ and 2$ are the values in the envelopes. Either you will gain 1$ or you will lose a 1$ by switching. The ultimate average is 0. If you happen to choose 1$, by the above logic, you will pick either 0.50$ or 2$ and the average is 1.25$. The real average is still 1.50$ and the gain is 1$. Similarly with 2$ in the chosen envelope. Now you have an average of 2.50$ with the above method. This is wrong once again. The real average is still 1.50$ and the loss is 1$. There is no advantage in swapping and flip-flopping around and again.
Now on a gain and loss analysis, a totally different subject than this, given the chance to gain twice your money or lose half, bet all bets on the doubling! It's the odds on a 50% loss or a 100% gain, and that sounds very profitable and at worst case worth the risk. Besides, you'll still have half of your money either way.
A thought provoking problem however you look at it.
Posted by Michael
on 2004-11-22 05:26:31