It is clear that if the contestant sticks with his first guess, he has 1/3 odds of being correct. If the contestant does not stick with the first guess, there are three equally likely possibilities: the car is not behind either remaining door, the car is behind the first remaining door, the car is behind the second remaining door. Since in each case the host eliminates one of the doors without the car, the single remaining door will have the car two-thirds of the time.