This answer is 1/2, not 2/3, and points out a very obvious flaw in probabilty: that is prespective in interpretting what the event is makes a difference. Consider the following.

If you have two doors, one with a car, another with the goat. You're asked to find the car. The probablilty is 50/50

car goat -> choose.. p = 50/50

(fine)

If you choose one and are given a chance to choose again, the probability is still 1/2

car goat -> choose one, think it over, then choose again. p = 1/2

(so far so good?)

Now let's say the peron is offered a third incorrect choice which he/she knows is incorrect. Since there is no way he/she will choose it, it will not affect the outcome so the probability is still 1/2

car goat known-goat -> choose one knowing that a door you will never pick is the wrong one, think it over, choose again.. p = 1/2

If you look really carefully, this is the Monty Hall problem.

Once Monty opens the one door with a goat, you're presented with a brand new problem which in no way depends on the previous choice. Like economic sunk costs, the previous decisions were become irrelevant.

To visualize this, imagine the person making the "switch, no switch choice" is different. This new person is told the following: one of these doors has a goat, the other one a car. A third door has a goat but really, you don't care about that. One last thing, a guy/girl before you made a guess which you don't know whether it is right or wrong. He/She picked one of the two doors you're given. His/Her guess was harder because I had more doors then. Anyhow he/she guessed door 'x' but I guess you probably don't care since you still don't know if it was right. Now please choose a door." Basically you can see thatn it the Monty Hall probelem isn't one of probabilistic saviness, rather simply one packed with irrelevant information, enough to fool the most ambitious mathematicians.

When Monty opens a door revealing one which is wrong. You pat yourself on the back for not having guessed that particular goat but that still does not change the nature of your new question. Now Monty says "there is a goat and a car behind on the two remaining doors. Which one do you think has the car, the one you chose originally(door A) or the other one(door B)?"

car goat -> choice.. p = 1/2

It doesn't take a lot of intelligence to see through this, it only takes the ability to see beyond your own arrogance.

The root of this problem is that too often, logicians and mathematicians treat randomsness as if it were logical: always following the rules and axioms it is supposed to. In the math world, that's fine. The real world however, where most probability is applied, it's not so simple. Weather predictions are wrong; polls don't tell you who will win the election; having a low city crime rate will not mean you can never get mugged. If people were able to recognize this simple fact, perhaps less time would be wasted fiddling around with numbers and losing money in casinos.