The Monty Hall Game

You are a contestant on a game show. Here are the rules of the game show:

There are 3 doors, and Monty Hall, the host of the game show, has made sure that there's a car behind one of these 3 doors, and man-eating goats behind the other two doors.

At the start of the game, you may choose any one of the three doors.

After you pick one of the 3 doors, Monty Hall opens one of the two doors that you did not pick, and shows you that there is a goat behind that door. (Monty Hall is not allowed to open your door, and he knows where the car is and will not open a door if the car is behind that door .)

Finally, Monty Hall asks you if you want to switch doors. In other words, you can either choose to stay with the door that you initially chose, or switch to the other door that’s still unopened.

You win if you ultimately choose the door with the car. If you wind up with a goat, it eats you.

Suppose you choose the first door at the start of the game.

Now suppose Monty shows you a goat behind Door 2.

What is the probability that the car is behind Door 3 ?

Note: So far, you only know what is behind Door 2.

Another note: You can find this on Brilliant in the Perplexing Probability course.