The Monty Hall Problem
The Monty Hall Problem is named after the host of the US TV show 'Let's Make a Deal' and is a fantastic example of how our intuition can often be wildly wrong when trying to calculate probability. In this article, we are going to look at what the problem is and the mathematics behind the correct solution.
Suppose you are the winning contestant on a quiz show, and for your grand prize, you are given the choice of three doors. Behind one of the doors is a brand-new car, while behind the other two are goats. You win whichever prize is behind your chosen door.
You choose a door, but the TV host asks you to wait for a moment. He then opens one of the other two doors to reveal a goat and gives you the option of switching doors. You can either stay with the door you originally chose or switch to the other remaining closed door. Should you switch?
Suppose you are the winning contestant on a quiz show, and for your grand prize, you are given the choice of three doors. Behind one of the doors is a brand-new car, while behind the other two are goats. You win whichever prize is behind your chosen door.
You choose a door, but the TV host asks you to wait for a moment. He then opens one of the other two doors to reveal a goat and gives you the option of switching doors. You can either stay with the door you originally chose or switch to the other remaining closed door. Should you switch?
Monty Hall (1921 - 2017)
Should you switch doors?
Intuition seems to suggest that it shouldn't matter whether you switch doors or not. There are two doors left; one has a car behind it, and the other has a goat, so you would think that it is a 50/50 choice either way. However, that isn't the case.
If you switch doors, you are actually twice as likely to win as if you didn't switch. This is so counter-intuitive that even many university professors of maths argued passionately against it when first faced with this problem.
Let's look at how it works.
If you switch doors, you are actually twice as likely to win as if you didn't switch. This is so counter-intuitive that even many university professors of maths argued passionately against it when first faced with this problem.
Let's look at how it works.
Why should we switch doors?
Look back at the picture at the top of the page. Suppose you pick door two. The TV host then opens a door to reveal a goat. He knows where the goats are, so the open door will always be a goat; he won't reveal the car by accident.
This leaves two doors, and we know that one has a car behind it and the other one has the other goat behind it. Therefore if we switch doors, we are guaranteed to switch prizes, either from car to goat or from goat to car.
You choose to switch doors. For the new door to have the car behind it, you need to have started off pointing at a goat door. If we can work out the probability of originally pointing at a goat, we, therefore, have the probability of the new door having a car behind it.
This leaves two doors, and we know that one has a car behind it and the other one has the other goat behind it. Therefore if we switch doors, we are guaranteed to switch prizes, either from car to goat or from goat to car.
You choose to switch doors. For the new door to have the car behind it, you need to have started off pointing at a goat door. If we can work out the probability of originally pointing at a goat, we, therefore, have the probability of the new door having a car behind it.
Monty Hall Problem Prizes
Matti Blume - Wikimedia Commons
The probability of starting on a goat
As there were three doors to choose from at the beginning and two of those doors had goats behind them, the probability of picking a goat with your first choice of door is 2/3.
This is the outcome that would lead to switching doors giving you the car, hence if you switch doors, the probability of winning the car is 2/3, twice as big as the probability of winning if you stick with your original choice (1/3). Difficult to believe, but true!
This is the outcome that would lead to switching doors giving you the car, hence if you switch doors, the probability of winning the car is 2/3, twice as big as the probability of winning if you stick with your original choice (1/3). Difficult to believe, but true!
Why does this work?
The thing to remember here is that even though you have ended up with only two closed doors, the host's choice of which door to open to reveal a goat was dependent upon your original choice of door; it is the probabilities of the original three doors that are important.
An alternative way of thinking about it
In case you are still not convinced, here is another way to look at the Monty Hall Problem.
There are three possible combinations behind the doors. Either the car is behind door 3, door 2, or door 1 and the goats fill up the remaining two places in each example.
There are three possible combinations behind the doors. Either the car is behind door 3, door 2, or door 1 and the goats fill up the remaining two places in each example.
Three options of car placement
Examples
In the picture above, we are looking at what could happen if your original choice was door 1 (shown by the black arrow). In the top row of the picture, you choose door 1, the host opens door 2 to reveal the other goat and so switching will take you to door 3 and the car.
In the second row, we have a similar example. You start on door 1, the host opens door 3 to reveal the other goat and you switch to door 2, again winning the car.
In the bottom row, however, you start off pointing at the car, the host then opens one of the two remaining doors and switching will take you to the other goat.
So if you start on door 1, there are three possible outcomes when switching, two of which lead to winning the car, hence the probability of switching giving you the car is 2/3.
It can be seen quite quickly that the same would happen if you originally chose doors 2 or 3. So regardless of which door you start on, there will always be a 2/3 chance of winning by switching. This then gives you an overall probability of winning by switching of 2/3.
In the second row, we have a similar example. You start on door 1, the host opens door 3 to reveal the other goat and you switch to door 2, again winning the car.
In the bottom row, however, you start off pointing at the car, the host then opens one of the two remaining doors and switching will take you to the other goat.
So if you start on door 1, there are three possible outcomes when switching, two of which lead to winning the car, hence the probability of switching giving you the car is 2/3.
It can be seen quite quickly that the same would happen if you originally chose doors 2 or 3. So regardless of which door you start on, there will always be a 2/3 chance of winning by switching. This then gives you an overall probability of winning by switching of 2/3.
Comments
What do you think? Don't forget to leave your comments below.