Correct!!!
The answers 2/3 and 1/2 are both correct, depending on when was the event measured from.
Further the one thing that is to be kept in mind is there should be no distinction between the goats( both are losses) to win the game if we start the event count after one of the goats is found then the probabilty is 1/2 cause the car is definitly behind any of the two unopened doors..
In case if we start to measure the event form the beginning then :
1.when all doors are closed: probabilty is 1/3
2.when one of the doors are opened the probabilty is 1-1/3=2/3
where 1 is the event of one door opened and a goat is found behind.