The probability that your initial choice did not have a car is indeed $2/3$.

Initial Misstep: After one door is opened, there are exactly two doors left, and one of them has a car. So the probability that the car is behind either door is $1/2$. This is incorrect because the host knows which door has a car and which door has a goat. The host always opens a door with a goat.

Correct Solution: The following table shows which Door the host might open. Assume that the car is behind Gate #1, and we randomly choose one door.

Initial choice:	Door #1	Door #2	Door #3
Reality	Car	Goat	Goat
Host opens:	#2 or #3	#3	#2
Remaining:	#3 or #2	#1	#1
Good to switch:	No	Yes	Yes

We see that at the end, the remaining unopened door is Door #1 if we start with Door #2 or Door #3. This means that in 2 out of 3 cases, we started with the incorrect Door (#2 or #3) and we got the option to switch with the correct door at the end (#1).

So you should switch to the other door, and win the car with a probability of $2/3$

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Generalization

The probability of being initially wrong is the same as the probability of being correct after switching. We can generalize this to $n$ doors. The probability of winning the game by switching after the host has opened $n-2$ doors is $(1 - 1/n)$

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Script

Not convinced? Simulate using this Colab Script