Monty Hall problem
The Monty Hall problem is a famous probability puzzle. The problem is based on a television game show from the United States, Let's Make a Deal. It is named for this show's host, Monty Hall.
In the problem, there are three doors. A car (prize of high value) is behind one door and goats (booby prizes of low value) are behind the other two doors. First, the player chooses a door but does not open it. Then the host opens a different door. The host knows what is behind every door, and always chooses a door with a goat behind it. If there are goats behind both other doors, one is chosen at random. The player must then choose: stick with the original door, or switch to the remaining door (the one the host did not open). The question is: will switching increase the chance the player will win the car?
Many people think that the car is equally likely to be behind either of the two doors that are still closed. Because of this, they think that switching doors has no effect on the chance of getting the car. The true answer is that changing choices increases the chances of getting the car from 1/3 (one out of three) to 2/3 (two out of three).
The key point of this puzzle is that the host will never open a door with the car behind it. By the rules of the game, the host must open a door with a goat. This gives the player useful information. The host shows which of the other two doors is the bad one by opening a goat door. This is why switching gives you a 2/3 (two out of three) chance of winning the car. When you switch, you are swapping one door for the best of the two doors not picked at first.
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The player has an equal chance of initially selecting the car, Goat A, or Goat B. Switching results in a win 2/3 of the time. | |||||||||||||||||||||||||||
These are the options:
1. (Lose) : If the player picks the car, then the host will show a goat. Then if the player changes their choice, they will get a goat.
2. (Win) : If the player picks a goat, then the host will show the other goat. Then if the player changes their choice, they will get a car.
3. (Win) : If the player picks the other goat, then the host will show the first goat. Then if the player changes their choice, they will get a car.
So, it is true that if the player switches then they will win the car two times out of three.
Other websites
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- Monty Hall Problem at Wolfram MathWorld
- Graphical Proof of the Monty Hall Problem
- A Monty Hall Simulator in Javascript Archived 2007-01-29 at the Wayback Machine
- Monty Hall Simulation Online
- Making the Monty Hall Obvious
- A tree-diagram of the Monty Hall problem under the Marilyn vos Savant assumptions Archived 2007-03-11 at the Wayback Machine
- The Game Show Problem Archived 2010-03-10 at the Wayback Machine