The Monty Hall Problem: Should You Switch Doors?
Monty Hall was a game show host of the show called "Let's Make a Deal", in which a participant was placed in front of three doors with no knowledge of what was behind them. Monty Hall would then tell the participant that two of the doors have goats behind them, while one has a brand new car. The participant would then choose a door to open.
The game takes a twist here. Instead of opening the door the participant pointed at, Monty Hall then opens one of the other doors, and always reveals a goat. Then, he asks the participant, "Would you like to stick with your door, or switch to the other unopened door?" Well, what do you think? Should they switch?
Many of you are probably thinking, "its a 50 50 chance that he gets it right, one door has a goat, one has a car, so he has the same chance of getting either one." Well, that's wrong.
I was pretty baffled when my brother first showed me this problem and walked out of my room laughing and calling me an idiot, even though he didn't get the correct answer his first time seeing the riddle. This game show problem became a popular statistics problem and is known worldwide today as the "Monty Hall Problem". In this article I'm going to show you why participants should always switch doors (basically a summary of a khan academy video).
There's essentially two scenarios: one where the participant does not switch doors, and one where they do.
If we look at the probability of winning when a participant doesn't switch doors, we can treat it as if the participant was never shown the door. Because the participant is not influenced by what he or she was shown, it makes no difference that they know one of the other doors has a goat behind it or not. Therefore the probability of winning is 1/3, and the probability of losing is 2/3.
In the second scenario, the probability of winning is different. In this situation, you can initially pick wrong twice. 2/3 of the time, you will pick a door that has a goat behind it. That means that if you are shown a door with a goat behind it, you should ALWAYS switch to the other door. If you always switch doors, the only losing situation is if you had initially picked the correct door and then switched to a door with a goat behind it. However, the likelihood of initially picking a door with a car behind it is only 1/3, while the probability of picking a goat is 2/3. In the scenario where one picks a goat on their initial guess (2/3 of the time), and if they are shown another goat, it is certain that the door that has neither been opened or chosen MUST have a car behind it. Considering these factors, it becomes evident that the probability of winning if one switches doors is 2/3.