The problem is a take on the classic probability problem known as the "Monty Hall Problem". Let's say Monty asks you to choose between N doors where N >=to 3. Behind one of the doors, there's a car and behind the remaining doors are goats. The probability that you chose the door with the car behind it =1/N. Now lets assume that you have initially chosen a door, but then before opening the door you chose, Monty opens N 2 empty doors of the N 1 doors that is left. This results in only 2 unopened doors, the one you initially picked and the remaining one. What is the probability

of you getting the door with the car if you switch your decision (e.g. P(winning if switch))?

Please compute this probability (e.g. you can use bayes theorem, total probability theorem, conditional probabilities, and such). Please show all work (e.g don't put P(something)=1 without showing work). I'd like things to be worked out completely.