If the host picked randomly then there would be a 1/6 chance that the show would end rather anticlimactically and you would be indifferent in switching. Now this 1/6 becomes your edge turning your 1/3 to 1/2 after the goat is revealed.
1/3 not 1/6. There are six outcomes of you picking a door and the host opening one of the remaining doors. In two of them you pick a goat the first time and then the host reveals the car.
EDIT: 1/3 not 1/6