I must say this is one of the best comments I have seen on HN. I don't know if this helps you meditate on the game or not, but this is how I walk though the logic.

1/3 of the time you picked correct: 100% of those worlds he shows a goat. 1/3 of the time you picked wrong and he shows a goat. 1/3 of the time you picked wrong and he shows a car.

So your overall odds are 1/3 of winning if you don't change AND 1/3 of the time you don't get the chance to change.

Thus, your overall chances of winning in this case is 1/3. But, if he shows you a goat your odds are bumped to 1/3: 1/3 or 50/50.