This is not the Two Envelopes Problem that I am familiar with, but still a good one overall. Say you have an adversary. The adversary has two envelopes and in each envelop writes a number. It can be any kind of number as long as its real. You get to open on of the envelopes and see the number that is written, and then you must guess if the other envelope contains a number larger or smaller than the one you opened. Is there any strategy that you can employ to raise your probability of winning to better than half? Think about the answer for a while and then click the link above to see his solution.

Quoted from Wikipedia:

Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice?

If you think you know the answer, follow the link to the wikipedia entry to find out for sure. But I will warn you, most people, even mathematically inclined individuals get this wrong the first time.

See also: The Two Envelopes Problem