Rather than going through this convoluted kinda similar problem, I find it easier to stick to the original one.

Get a piece of paper. Draw all possible outcomes, 9 total. ( Car is behind door 1 you pick 1, Car is behind door 1 you pick 2...). 3 of the 9 result in success.

Now draw the outcomes again but switch every time. 6 out of 9 outcomes are a success.