Math Puzzle No. 2: The Monty Hall Problem
COME ON DOWWWWNNNN!!!!!! You’ve been chosen to be a contestant on Let’s Make a Deal! You’re now standing in front of three doors -- behind one of these doors is a car, and behind each of the other two is a goat.
Monty, the host, says, “Pick a door! If it has the car behind it, you’ll drive it home today!” You choose door #1, and feel really good about your choice. The host then opens door #3, which has a goat behind it, and asks “Do you want to switch doors?”
Question: What is the probability that you will win the car now that you know there was a goat behind door #3? Should you switch?
Assumptions: Before the show, the car was equally likely to be placed behind any of the three doors. Once you make your selection of door #1, Monty will choose to reveal what is behind one of the other two doors, but will definitely not show you where the car is.
WRONG!
In 1975, a statistics professor, Steve Selvin, wrote a letter to the American Statistician where he proved that a contestant of the show can increase their chances of winning the car by switching doors!
If you’re thinking to yourself, What?!? This seems crazy!, you’re not alone. Following this letter, Steve himself received many letters from people criticizing his solution. It just didn’t seem right! This problem has since been referred to as The Monty Hall Problem -- named after the game show’s producer and original host, and to this day, it continues to drop the jaws of those learning statistics and probability.
Let’s Experiment:
So why should you switch? We would first like to encourage you to find a friend, and play this game about 30 times.
Choose a number between 1, 2 or 3 (this will signify the door number that the car is behind). You can write down your choice, put a sticky note on one side of an index card, or otherwise mark down your choice. Keep your number secret for now.
Ask a friend to choose a number: 1, 2, or 3.
Reveal a number that has not yet been chosen by either one of you (Since the number that you chose is the door that “hides the car”, this will signify “revealing a goat” behind one of the two doors that hides a goat.).
In the following table, mark down whether your friend would win or lose by switching;
In the first row, mark down a W if they would win by choosing to switch numbers (or “doors”), and an L if they would lose by switching.
Then, in the next row, mark down a W if they would win by not switching, and an L if they would lose by not switching.
So, what do you think? Should you switch?
The Mathematics Behind This Choice:
In statistics, we call the set of all possible outcomes of a situation a sample space. Initially, the sample space has three items, since there are three possibilities: 1-- the car is behind door one, 2-- the car is behind door two, and 3-- the car is behind door three. The Monty Hall game operates under the assumption that the host randomly chooses which door the car is behind -- thus each of these outcomes has an equal probability (⅓) of occurring.
Initially, no matter which door you choose, there is a one in three chance that you are correct. Assume for example that you choose door number 1. There are now three, equally likely, events in the sample space:
The car is behind door 1, and you are win!
The car is behind door 2, and you lose.
The car is behind door 3, and you lose.
But, the game does not end here! In the first scenario, the host will open either door 2 or door 3. In either case, you lose if you switch, but win if you don’t switch. In the second scenario, the host must open door 3 (because the host cannot open the door you have chosen, and cannot reveal the car). In this case, you win if you switch your choice to door 2, and you lose if you keep your original choice. Similarly, in the third scenario, the host must open door 2. Again, you win if you switch, and you lose if you keep your original choice.
Remember, each of the three scenarios above has an equal chance of occurring. Thus, if you originally choose door 1, and then switch doors (both choices you are in control over), there are three equally likely events in the sample space:
The car is behind door 1, the host either door 2 or 3, you switch (You lose).
The car is behind door 2, the host reveals the goat behind door 3, you switch (You win).
The car is behind door 3, the host reveals the goat behind door 2, you switch (You win).
No matter which door you originally choose, if you switch doors, you will win 2 out of 3 times! In fact, you win whenever you choose a door hiding a goat at the first stage of the game.
On the contrary, if you originally choose door 1, and do not switch doors (both choices you are in control over), there are three equally likely events in the sample space:
The car is behind door 1, the host either door 2 or 3, you do not switch (You win).
The car is behind door 2, the host reveals the goat behind door 3, you do not switch (You lose).
The car is behind door 3, the host reveals the goat behind door 2, you do not switch (You lose).
If you do not switch doors, then you win only in the event that you correctly choose the door hiding a car at the first stage of the game.
This can be summarized with the following table:
Another way to visualize this (pay close attention to the fractions):
Upon making your initial choice, there is a ⅓ chance that you get goat A, a ⅓ chance that you get goat B, and a ⅓ chance that you get the car.
If you chose goat A, then the host opens the door hiding goat B with probability 1. Thus there is a ⅓ chance that you choose goat A and the host reveals goat B. Similarly, there is a ⅓ chance that you choose goat B and the host reveals goat A. Now, if you chose the door hiding the car, then there is a ½ chance the host will show you goat A and a ½ chance that the host will show you goat B. Thus, there is a ⅙ chance that you initially choose the car and the host reveals goat A, and a ⅙ chance that you initially choose the car and the host reveals goat B.
If you choose to keep your original door, then you get goat A ⅓ of the time, goat B ⅓ of the time, and a car ⅓ of the time. If you switch doors, you get a car ⅓ + ⅓ = ⅔ of the time (in the cases that you originally chose a goat), and a goat ⅓ of the time (the case that you originally chose the car).
Why does our intuition fool us?
That’s a tough question! When we initially think about this problem, we consider these events as independent, random events. But, when the host reveals a door, the choice of door depends on both the location of the car and our original choice. Our intuition does not consider the fact that we are obtaining information when the host reveals a goat behind one of the doors. But, if we know the mathematics behind the Monty Hall Problem, and use the information given to us by the host, we can in fact DOUBLE our chances of winning a car!
Because our choice determines what the host has to do to make the sample space smaller, changing our choice after we get more information is a good idea much -- in fact, exactly ⅔ -- of the time!
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