If there was no influence (i.e. the game restarted if the hose revealed the car by mistake) then there would be two independent choices. 1:3 1:2. with stick or change both having a 50/50 choice.
It doesn't matter in that case. You've accidentally arrived at the exact same situation. You still have a 2:1 better chance switching at that stage.I've been trying to get my head around that point.
Say, the game was played, but the host doesn't know where the car is. The contestant chooses a door, and then the host opens a door which (by chance) reveals a booby prize. What would the odds then be on sticking or changing to win the car?