Tags Math
This paradox is very well known, but today I gave it a hard thinking and talked with a friend about it. It's called the Two Envelopes paradox.

Suppose I offer you a game: I present you with two envelopes, and tell you that one of them contains twice as much money as the other, and ask you to pick one. You pick one, and discover \$1000 dollars in it. Then, I offer you to change your mind and pick the other one. Will you do so ?

Where's the paradox ? Well, suppose I do the following calculation: OK, the other envelope has \$2000 with probability 1/2 and \$500 with probability 1/2. So, if I change, the expectancy of the sum I get is: 2000/2 + 500/2 = \$1250, meaning that I should switch ! Phew, lucky for me that I know some probability !

But hey, hey, wait a second. Does it matter what sum is in the envelope I opened ? Say it's N dollars. So, when I see N dollars, should I switch ? According to that calculation, yes, because if I switch, on the average I'll get 2N/2 + (N/2)/2 = 5N/4, which is more than N. So, whatever the sum is, I should switch ! But then, why open the envelope at all, if I will switch any way ? Why not just think what envelope I'll open, and then immediately switch ? Sounds wacky yet ? Say I picked an envelope, and without opening it, switched. Doesn't it lead me to the initial state ? Why shouldn't I switch again, if the same calculation works in this case also ?

This is the paradox... Although it seems pretty simple, its solution isn't... If fact, there is no one accepted solution. Many scholars proposed solutions, some profoundly wrong. People wrote scientific articles on this paradox... Can't blame them, it does look mind-boggling, and the more so once you start thinking seriously about it.