Suppose that you are a contestant on a game show and you have to choose between

n closed doors. Whatever is behind that door is your prize. Behind one door is a brand new car

(which is what you would like to have) and behind the other n - 1 doors are goats. The host of

the show, Monty, knows where the car and goats are located, but you, being the contestant, does

not. Suppose you pick a door at random. Before Monty opens the door you chose, he randomly

opens k doors such that each of the k doors have a goat behind it, where 0 k n - 2. (Note

that he never opens the door with the car during this process). He then asks you if you would

like to switch your door to any of the remaining doors. Should you choose to switch your door,

you randomly select any of the remaining doors. If you decide to switch your door, what is the

probability that you win the car?