The following puzzle has an answer of “Yes” but the one posted above has an answer of “There is not enough information to make that determination.”
“You’re on a game show, and you’re given the choice of three doors. The host explains the game by saying that you will pick a door, he will then show you a goat behind one of the other doors, and then you will have an opportunity to switch… (Etc.)”

The only way the answer to this problem is “yes” is if your option to switch doors is a rule of the game.

If the host has been told by the producers to try to not give you the car, then the reason he offers you the choice to switch could be because you have chosen the door with the car. Because you do not know his motivation, you cannot say whether you should switch, given the way the problem is stated.

think about it another way: instead of 3 doors, there are 1000 of them. you pick one, the host opens 998. what do you do?

the correct answer is: switch, you just got a car!

Read the wikipedia article it makes far more sense than the given solution, however, I’ll try to summarize. Our choice at the beginning is either a car, goat, or a goat each with a 1/3 probability. The only way that we will lose if we switch is if we pick the car from the start. Since there is a 67% chance we will pick a goat at the start when we switch after the first door reveals a goat we will have chosen a car

I believe that this is not merely a statistical problem, but also a psychological one. Because, mind you, if their are two goats, regardless of your selection, they can show you the other one. This really means that, if they are always going to expose a bad choice after your choice, you really only have a 50% chance, based on the original choice and the one they don’t show. This is much more an opinion-type logic than a factual-type logic.

The solution claims 3 possibilities, but there are 4.

1. The player has picked the door with the car. The host has shown Goat A.
2. The player has picked the door with the car. The host has shown Goat B.
3. The player originally picked the door hiding Goat A. The game host shows Goat B.
4. The player originally picked the door hiding Goat B. The game shows Goat A.

The answer is correct, but the solution is not.

“By switching, you effectively ‘open’ 2/3 doors.”
Nuh-uh.

If you switch, 1 door has been opened.
If you don’t switch, 1 door has been opened.

If you switch, 1 door will be opened next
If you don’t switch, 1 door will be opened next.

Both the 2 unknown doors now have a 1/2 chance of win, not determined by the opened door.

“Always account for variable change!” -Kevin Spacey, 21