The Monty Hall Problem

One problem that I have read about in several books involving probability is somewhat popular and is known as the Monty Hall problem. It caused much debate in the 1990s and is described by the following:
    
"The set-up is a standard game show in which the contestant is trying to win a prize secreted behind one of three closed doors. Behind two of the doors is an amusing but essentially worthless prize, like a sheep, while the third conceals something desirable, like a Mercedes. When the contestant chooses a door, the game-show host quickly opens one of the two remaining doors and reveals a booby prize. The host then offers the contestant a choice: Either stay with the original pick or switch to the other closed door. That's the situation. The question: Is it in the contestant's best interest to switch or does it not matter?" ( Strange Universe by Bob Berman). Note: Monty Hall, the host, knows which door the desirable prize is behind.

The solution is somewhat counter-intuitive and was highly debated among mathematicians. My first thought was that it didn't matter and that each door has the same 1/3 chance in containing the Mercedes. However it is in the contestant's best interest to switch to the other door. Why? Switching gives the contestant a 2/3 chance of winning whereas keeping the original door only gives a 1/3 chance. Let me explain. Originally there is a 1/3 chance that the contestant's first choice, lets say Door A, is the winning door. There is a 2/3 chance that either Door B or Door C is the correct door.
So Door A is set 1 and Door B and C are set 2. The odds say that the prize is more likely to be in set 2 however the contestant cannot pick 'set 2' they must choose a specific door. Then Monty Hall opens Door B in set 2 which he knows does not contain the prize. This leaves Door C as the only door left in set 2. Since there is still a 1/3 chance of the winning door being in set 1 and a 2/3 chance in the door being in set 2, the contestant is more likely to win by switching to the other unopened door, in this case Door C. This solution has been tested and supported with computer simulations of a similar scenario.

Let me know in the comments if I explained it correctly. Have you heard of the Monty Hall problem? Do you agree with the solution? Does it even make sense?

The Paradoxicality