Here is a riddle that I've heard long time ago:
Say you were on a TV game show with a gazilionare. The gazillionare owns infinitely much money. He writes down two numbers on two pieces of paper; one on each. One of the numbers are twice as large as the other. Then he puts the papers inside two envelopes: one for each paper. Of course the two envelopes are identical. He then writes an "A" on one of the envelopes, chosen at random, determined by a coin toss. On the other he writes a "B".
Now he askes you to choose one of the envelopes. You win an amount of money equivalent to the the number written on the piece of paper. On the piece of paper inside the envelope it reads "X".
Now you've got another choice to make: switch envelope, or go for the chosen one. However, if you read the number inside the other envelope, you have to choose it: That is, you can't switch back again.
So what should you do? Switch envelope or keep it? At what amount of money should you stop switching envelopes?
Since there are only two envelopes, as an example, let's assume that the number in the first envelope is $100.00.
Now, the number in the second envelope will be either $50.00 or $200.00.
The probability that it will be $50.00 is ½. Also, the probability that it will be $200.00 is ½.
From the $100.00 you now have, there is a ½ probability of losing $50.00 (going from $100.00 down to $50.00) and there is a ½ probability of gaining $100.00 (going from $100.00 up to $200.00).
So, the expected value is: ½(-$50.00) + ½(+$100.00) = -$25.00 + $50.00 = +$25.00.
Therefore, switch! (The same analysis works for any amount in the first envlope.)
That's my mathematical answer.
My psychological answer is this:
-- Obviously, you would sooner gain than lose; but gains don't necessarily balance losses; that is, it takes more than one gain to offset just one loss.
-- If the number in the first envelope is sufficiently large so that losing one-half of it would cause you to be mad at yourself, take the first envelope and don't even check to see what's in the second envelope ...
I agree, but isn't it weard that one should switch no matter what? :) I mean, is it really necessary to peak in the first envelope? One would switch anyway?
Maybe one should choose the other one from the start? (Of course that doesn't work either, but you see my point)