Let’s Make a Deal is a long-running TV game show that debuted in 1963 and is still on the air today. The early versions of the show featured contestants who made bargains with the show’s co-creator and longtime host, Monty Hall. If the contestants were lucky and skilled at making deals, they could win cash and other prizes, but if they weren’t, they received undesirable “zonk” prizes such as goats instead. Who would have known that decades after the show premiered, a probability puzzle loosely based on it would stir up so much controversy in academic and intellectual circles. We will examine this intriguing puzzle today, in this episode of The Infographics Show, “The Monty Hall Problem.”
The probability puzzle, also known as the Monty Hall Problem, is similar to the “three curtain” game of Let’s Make a Deal, except that it features three doors in place of the curtains. One door has a car hidden behind it, while a goat is hidden behind each of the other two doors. You pick the door that you think will win you the car. After you select a door, Monty Hall opens one of the other two doors and reveals a goat behind it. At this point, he gives you a choice: you can stay with the door that you selected, or you can select the other door instead. Should you stay with your original choice, or should you switch?
On the surface, this seems like a simple question, but the three main solutions to this problem have been a source of debate for many years.
1. The “Common Sense” Solution
If you know that a goat is behind one of the three doors, then there are only two doors left to choose from. A goat is behind one of the doors and the car is behind the second remaining door, so your odds of picking the door with the car is 50-50. This is the solution that is the easiest for most people to understand, but now many experts think it is wrong.
2. Marilyn vos Savant’s Solution
Marilyn vos Savant is known as the woman with the world’s highest recorded IQ. She has worked for Parade magazine since 1986. In her column called “Ask Marilyn,” she responds to a wide variety of intellectually challenging questions from readers. On her website, she devotes an entire page to the “Game Show Problem” that a reader sent to her in 1990:
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?
This was her solution to the reader’s version of the Monty Hall Problem:
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?
Some additional explanation will further clarify her solution. Because there are three doors, each one of them has a 1/3 probability of having the car behind it. The opening of a door that reveals a goat does not cause the probability to be equally divided between the two remaining doors. As vos Savant puts it, “The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door.” Instead, the probability of the opened “goat” door having the car behind it is transferred only to the door that you do not select, which gives the unselected door a 2/3 probability that it has the car behind it.
This solution revealed the Monty Hall Paradox: the obvious “common sense” solution to the Monty Hall Problem is not the correct one. This paradox upset a lot of people because, as one reporter put it, “it’s an example of math completely contradicting my gut instinct.” Marilyn vos Savant received widespread criticism from shocked Ph.D.’s across the country. In their letters, they made disparaging comments to her like “You blew it!”and “Your answer is clearly at odds with the truth.”
But vos Savant did not back down. She defended her answer in additional columns. She presented her critics with the following table showing what she called the “benefits of switching.” The first three games show that you win 2/3 of the time if you switch, while the second three games show that you win 1/3 of the time if you stay. However, these results are based on the assumption that the “host always opens a loser.”
Door 1 Door 2 Door 3 RESULT
|GAME 1||AUTO||GOAT||GOAT||Switch and you lose.|
|GAME 2||GOAT||AUTO||GOAT||Switch and you win.|
|GAME 3||GOAT||GOAT||AUTO||Switch and you win.|
|GAME 4||AUTO||GOAT||GOAT||Stay and you win.|
|GAME 5||GOAT||AUTO||GOAT||Stay and you lose.|
|GAME 6||GOAT||GOAT||AUTO||Stay and you lose.|
In one of her columns, she also urged students in math classes throughout the country to perform another probability experiment that would test her “thinking” on the Monty Hall Problem. It involved three paper cups, a penny, and a die.
This approach worked. After conducting the experiment, many people became convinced that her solution was the correct one. However, some people still found fault with it. They thought that vos Savant inaccurately assumed that Monty Hall would follow the same pattern of behavior like a robot, which was not a realistic portrayal of human behavior. This leads us to a third solution.
3. The “It Depends on Monty” Solution
Proponents of this solution argue that psychology and math work hand in hand in the Monty Hall Problem. The motives of the host influence the assumptions made in the problem, and these assumptions in turn affect the probability of choosing the door with the car behind it. Here is an example of this line of thinking from a university professor:
My best advice is to look Monty in the eye and see if you can work out if he is trying to con you or not, or maybe if he is genuinely trying to give you another chance. Think about how many cars he has given away so far and assess whether Monty might be trying to encourage more winners or more losers. How long is there to go before the end of the game, and is Monty trying to spin it out or bring it to a halt? When you have decided that you can do all the math [sic].
The major problem with this solution is that having full knowledge of someone else’s motives is not possible. Even someone like James Bond who is keen at “reading” people is guessing about the motives of another person. It adds a new layer of complexity to the Monty Hall Problem that is hard to figure out: How will the probability that you will guess wrong about Monty Hall’s motives impact the probability that you will choose the right door to win the car?
The “It Depends on Monty” solution is more likely to land you in an intellectual impasse than in a new car. Even some of the supporters for this solution seem stumped. One article that references the Monty Hall Problem asserts that “the probabilities all depend on the rules for Monty’s behavior,” and at the same time asks, “But how in the real world could you know what the rules governing Monty’s behaviour are?”
So, do you know of another puzzle that has a counterintuitive solution like the Monty Hall Paradox? Let us know in the comments! Also, be sure to check out our other video called This Will Happen in 60 Seconds! Thanks for watching, and, as always, don’t forget to like, share, and subscribe. See you next time!
Let’s Make a Deal Show
Marilyn vos Savant