Discrete Probability Theory

A contestant chooses one of three doors; behind one of the doors is a car, and behind the other two are goats. After the contestant chooses a door, the host opens one of the other two doors that hides a goat. (Because there are two goats, the host can open a door that hides a goat no matter which door the contestant first chooses.) The host then gives the contestant the option of abandoning the chosen door in favor of the still-closed, unchosen door. If the contestant switches to the unchosen, unopened door. What is the probability of winning the car?

My best guess is:

(1/3) * (1/2) = 1/6

This is a well known problem: Monty Hall Problem

Thanks a bunch. I thought this problem would have a simple solution.