Yes, the whole point is that the car doesn't move. When you initially chose, you had a 1-in-3 chance of correctly choosing the door with the car. The host is no

Yes, the whole point is that the car doesn't move. When you initially chose, you had a 1-in-3 chance of correctly choosing the door with the car. The host is now offering you to switch over to, essentially, "all the doors you didn't choose", which means 1-(1/3). As you state, you've only got one opportunity to play, and the car itself hasn't moved, but you can switch your choice. Stick with the initial 1-in-3 chance, or go to the other side, i.e. 2-in-3 chance?