Statistics Sample Essay: The Monty Hall Problem

“Suppose that instead of three doors, there are four doors in the Monty Hall problem. What is the probability that you will win by not changing the door?

Statistics Sample Essay The Monty Hall Problem

Monty Hall dilemma is one of the puzzles in probability theory. It was named after a well-known producer, sportscaster and host of the famous TV game.

Three Doors’ Choice

The original version is follows: a player has to select one of three doors offered by the show host. There is a car (the prize) behind one door while the remaining two doors hide two goats. A player picks one of the doors (No. 1, for example). The host then opens another door with the goat (No. 3). Than he asks the player whether he wants to change his choice and pick a door No. 2. What are the chances to win a car if the player accepts the offer and changes his choice?

Extended Version: Four Doors Instead of Three

In this case we have four (or n-number) doors, one door with a car and the other three doors with goats. We choose one door, and then one of the doors that has a goat is revealed. The host asks us if we want to change our choice. Should we change the door we chose or remain with our choice? The goal is to find out the possibility of getting the car.


The answer to this question may be shown with math statistics and equations. So, we have four doors (n=4), the possibility of winning the car is 1/4 . The host opens one (name it x) door, where 0 <= x <= n-2. In cases when x=1, n-1/n-x-1>1, the probability of finding the car on the second guess is more than 1/n.

Two doors remain, one of them with the prize (the car we want to win), 3 * 1/4 is the remaining probability. The final equation is is 3/4 *1/2 = 3/8.

If you change your choice your chance will be increased. If you don’t change your choice your chance is 1/4 . Therefore, switching is optimal.

Having N doors is a similar case, which shows that switching is preferable. So, if players would solve this puzzle with an infinite amount of doors and change their selection they would definitely win.


Monty Hall Problem. (n.d.). Retrieved May 24, 2016, from
Statistical Ideas (n.d.). Retrieved May 24, 2016 from”

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