The answer is counter-intuitive, but its correct. Here it is:
You actually double your chances by switching.
and heres why
There are 3 things that could happen:
1. You could choose the BMW. The host then shows you either of the two goats. Switching will lose.
2. You could choose the first goat. The host then shows you the other goat. Switching will win the car.
3. You could choose the second goat. The host then shows you the other goat. Switching will win the car.
So, initially, you have a 1/3 chance of getting the car.
If you switch after one of the goats has been eliminated, your chances increase to 2/3.
another way to look at it is:
since the host is showing you one of the goats after your first pick, that means when you switch, you are definitely changing from goat to car or car to goat. That means if you think you picked a goat on the first try, you should switch. well, you had a 2/3 chance of picking a goat on the first try, and only a 1/3 chance of picking a car. so switch. always switch.
Some people theorize that the main problem most people have with this puzzle is not mathematic, but psychological. The human strives to be in harmony with his past decisions. If someone makes a choice and then is shown that the choice he has made might require rethinking, he has a tendency to stand by his original decision even in the face of new contradictory evidence. Its actually a very important function of the brain that helps streamline thinking. If you've made a choice about a particular subject, then it would be inefficient to rethink it every time that choice arises, so it serves a valuable role in the brain. although it sometimes goes awry.