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 ...
