Statistical Consultants Ltd

The Monty Hall Problem

Statistical Theory, Probability, Psychology

The Monty Hall problem is a simple probability puzzle that has managed to fool many people, including those highly educated in fields that rely heavily on mathematics.

During the American game show, Let’s Make a Deal, there was a game where the host (Monty Hall) would show the contestant three doors. Only one of the doors would have a prize behind it. The other two would have booby prizes e.g. a goat. The contestant would pick one of the doors. Monty would then pick another door and open it to reveal one of the booby prizes. He would then ask the contestant if they want to stay with the door they picked, or change to the other unopened door. Monty would then open the door of choice, and if the prize is behind that door, the contestant wins it.

Many people falsely assume that it doesn’t matter whether or not the contestant changes doors. In reality, the contestant would be more likely to win the prize if they change doors. The reason why, is because the first door Monty opens will never be the prize.

For example, let’s say the contestant picks door 1.

If the prize is behind door 1, Monty would open either door 2 or 3.
If the prize is behind door 2, Monty would definitely open door 3.
If the prize is behind door 3, Monty would definitely open door 2.

Each of these three scenarios would be equally likely, yet for 2/3 of those scenarios, the contestant would be better off switching doors.