The Monty Hall Problem

By Becca Smithers

In 1963 a game show called “Let’s Make a Deal” surfaced in America, and was hosted by a man called Monty Hall. One of the sections of the show has caused endless mathematical debate due to it’s perplexing probability, and is often referred to as the “Monty Hall Problem”.

Is it behind door number one…?

The contestant on the show would be shown three doors and was asked to pick one. The prize behind one of the doors was a red sports car, and behind the other two doors were goats. After the contestant has selected their door, Monty Hall would then open one of the remaining two doors to reveal a goat. The contestant is then given the choice of sticking with their original door, or swapping to the other closed door. What should they do?

Stick or Switch?

When the contestant makes their first choice, there is a 1/3 chance that their door will have the car. Monty then reveals a goat behind one of the other doors, meaning that the car is either behind the contestant’s chosen door, or behind the remaining closed door. This seems like the probability of winning has gone from 1/3 to 1/2

However, this is not the case. The probability is still out of three even after the goat is revealed.

When Monty reveals the goat, the contestant has a 1/3 chance of winning the car if they stick, but they have a 2/3 chance of winning if they switch.

The choices

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There are three ways the game can play out:

– The contestant chooses door number one. Monty opens door number three. Contestant should switch to win the car.

– The contestant chooses door number two. Monty opens either door number one or door number three. Contestant should stick to win.

– The contestant chooses door number three. Monty opens door number one. Contestant should switch to win.

2/3 times you should win if you switch your choice, 1/3 times you should win if you stick. You have a greater chance of winning by switching.

For a more visual explanation, watch the video below!




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