meteomonk said:
Would the odds not go from 33.3% to 50% after being revealed what was behind door 3? why on earth would changing his answer give him 66% odds?
The key that you're missing here is that this problem is not a strict probability problem. It's a conditional probability problem, because the contestant has more information available to him than someone who walked in off the street.
As a matter of fact, this is a good place to make this point: if a stranger walked in off the street after door #3 had been opened (and had no idea which door had been originally chosen),
that stranger would have a 50% chance of guessing correctly.
The contestant can do better than 50% because he has information that the stranger didn't: the contestant selected door #1, and whether or not the car was behind door #1, door #1 could not have been opened.
The question now is not "you have two random doors, which one has the car" - it is "given that you chose door #1 and Monty opened door #3, which one has the car".
Play out each scenario.
(1) Car is behind door #1.
=> Contestant chooses door #1, Monty opens either door #2 or door #3.
=> Contestant wins by staying, loses by switching.
(2) Car is behind door #2.
=> Contestant chooses door #1, Monty is forced to open door #3.
=> Contestant loses by staying, wins by switching.
(3) Car is behind door #3.
=> Contestant chooses door #1, Monty is forced to open door #2.
=> Contestant loses by staying, wins by switching.
The contestant has a better chance than blindly guessing because he forced Monty's actions in two of the three scenarios.