Car Goat Problem

Six or seven years ago, I took a philosophy class in which the professor introduced me to the “Car Goat Problem” (also known as the “Monty Hall problem“). The problem is this:

Imagine you are on a game show and there are three doors in front of you. Behind one of the doors is a car and behind the other two doors are goats. The game show hosts asks you to choose one of the doors (the one you believe has the car). Then, the host opens one of the other doors, which has a goat. The host then gives you an opportunity to change your mind. You win whatever is behind the door your choose. Should you switch doors or should you stay with your initial door? Or does it not matter?

So, what do you think? My solution is below.

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