The Monty Hall Problem: Your Intuition Can Lie!
It can be challenging to see the truth. Even when someone shows us the correct answer, sometimes we stubbornly cling to a mistake. The Monty Hall Problem offers an entertaining way for almost everyone to experience a strong intuition that happens to be wrong.
Back in the 1960s, Monty Hall hosted the long-running television game show Let's Make a Deal. In part of the show, Monty played a game that exploited an intuition that almost everyone gets wrong. He would show the contestant three doors and explain that behind one door was a car and behind the other two were goats. After introducing the game, Monty asked the contestant to pick a door. However, he did not open the door the contestant had selected. Instead, Monty would open one of the other two doors and reveal a goat. The contestant was left facing two closed doors, and she knew the car was behind one of them. At this point, Monty would ask her if she wanted to stay with the door she had chosen or switch. To add more drama, sometimes Monty would offer the contestant money to change her choice. He knew if he offered the contestant money to change, she would think he was trying to tempt her away from the door with the car. The suspense built up to the moment Monty opened the chosen door. If he had not tried to lure the contestant into switching, almost everyone watching thought there was an equal chance the car was behind either door. After all, there were only two doors, and therefore it had to be 50 50. However, this intuition is wrong. It is better to switch. If the contestant swaps to the other door, she has a two-in-three chance of winning.
Monty Hall's game was entertaining, but eventually, he dropped it from the show. The story would have ended there if not for a letter sent in 1990 to Parade magazine. Years before, the ten-year-old Marilyn vos Savant took two intelligence tests and scored the same as a 23-year-old. Based on this score, The Guinness Book of Records made her famous by listing her as having the 'World's Highest IQ.' Later, Parade magazine wrote a profile on her, which created so much interest that they saw an opportunity for a new Sunday column. The idea was for readers to send Marilyn' The cleverest woman in the world' puzzles or questions, and she would answer them in her column. In 1990, Craig Whitaker wrote to Marilyn, asking, "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?" Marilyn's answer was short. She wrote, "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 number 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 number 777,777. You'd switch to that door pretty fast, wouldn't you?" Craig's question and Marilyn's answer appear innocuous. However, they triggered a storm of protest. Within months Marilyn had received more than 10,000 letters. Almost all told her she was wrong. Here are just two examples that give you an idea of Marilyn's postbag:
"You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I'll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don't need the world's highest IQ propagating more. Shame!
Scott Smith, Ph.D.
University of Florida"
"Since you seem to enjoy coming straight to the point, I'll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and in the future being more careful.
Robert Sachs, Ph.D.
George Mason University"
Marilyn responded to the criticism by saying: "Good Heavens! With so much learned opposition, I'll bet this one is going to keep math classes all over the country busy on Monday. My original answer is correct. But first, let me explain why your answer is wrong. The winning chances of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door. To illustrate this, let's say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The chances that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the chances on the shell under your finger." Marilyn finished her response by presenting a table showing all the game's possible outcomes. However, the expanded explanation did not make any difference. The letters continued to pour in. If anything, she was getting even more mail. Marilyn counted the letters and found 92 percent of the general public's mail, and 65 percent of university respondents thought she was wrong.
Marilyn changed tactics. Rather than searching for a more compelling mathematical explanation, Marilyn invited math classes nationwide to play the Monty Hall game. She suggested they label three paper cups one, two, and three and randomly hide a penny under one of them. The students could then play the game and experience the game's dilemma. However, Marilyn went further. She asked them to test whether her answer was correct. To do this, they had to play the game 100 times, following one strategy and then the same for the other. At the end of the experiment, the class could total up how many times stick or switch won. Putting the two strategies to the test proved popular. In a subsequent column, Marilyn wrote, "We've received thousands of letters, and of the people who performed the experiment by hand as described, the results are close to unanimous: You win twice as often when you change doors. Nearly 100 percent of those readers now believe it pays to switch."
The storm of controversy about Marilyn's answers caught the attention of The New York Times newspaper. A journalist interviewed Monty Hall and asked about the Parade magazine question. Monty immediately focused on the psychological aspects of the game. In the interview, he described how sometimes he deliberately tried to alter the contestant's mindset by offering them cash to stick with their original choice. Hall explained the psychology behind the game: "[After I opened a door with a goat], they'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered...The higher I got, the more [they] thought the car was behind [the other door]. I wanted to con [them] into switching there. That's the kind of thing I can do when I'm in control of the game. You may think you have probability going for you when you follow the answer in her column, but there's the psychological factor to consider." At the center of Monty's explanation is the contestant's belief that the odds were one-in-two. Platt and Watkins (1997) measured what percentage of the population had the same intuition. They found that over 93% of respondents believed the chance of winning was 50 50.
Craig Whitaker's question to Marilyn was not the first of its kind. In 1975, Steve Selvin sent a puzzle to The American Statistician magazine. In a later letter, Selvin explained that he had based the puzzle on the television game show Let's Make A Deal, and in honor of the show's host, he called it the Monty Hall Problem. After Parade magazine made the puzzle notorious, people started asking why so many got the answer wrong. One area they studied was the wording of the problem. They found that when people thought about the puzzle, they made assumptions implied but not stated in the language. If you change the premises, the correct answer changes. However, this is a subtle distinction. It does not explain why Marilyn received so many letters because most correspondents applied the same assumptions as Marilyn.
Not only is the Monty Hall intuition common, but it is also firmly held. Marilyn experienced this. Her logical arguments and mathematical proofs did not convince people. Instead, her answers enraged them. The intuition was so strong that people were blind to the fact that they were wrong. Their stubbornness led to some unflattering comparisons between university PhDs and pigeons. Walter T. Herbranson and Julia Schroeder of Whitman College wrote a psychology paper titled, "Are birds smarter than mathematicians?" They had run a series of experiments to compare how pigeons and humans learned the best Monty Hall strategy. What they found was embarrassing for us humans. In short, the pigeons quickly found the best approach was to switch, whereas "humans failed to adopt optimal strategies, even with extensive training." Now there was light-hearted proof that birds are, in fact, smarter than mathematicians.
Herbranson's pigeon experiments showed how reluctant people were to give up their Monty Hall intuition. However, we can be more intelligent than pigeons. We can overcome our intuition. All we have to do is externalize. There is a big difference between thinking about a problem and physically doing the experiment. When you internalize, your intuition has nothing challenging it; you rely on your mental processes. However, when you externalize, you challenge your thinking processes. Marilyn did this when she asked school children to put her answer to the test. The students did not just sit at their desks and think about the problem. They got up, did the paper cup experiment, and saw the results. The teachers were so impressed by what happened they wrote to Marilyn describing this mental shift. They reported that initially, the students had the same 50 50 intuition. However, doing Marilyn's experiment and seeing the switch strategy win twice as often as the stick strategy convinced them Marilyn's answer was correct. The experience of doing the paper cup trial challenged the students' intuition. In effect, they let the experiment decide the truth. In doing this, they took the ultimate step in externalizing.
Monty Hall used his game because it surprised and entertained his audience. They loved the format, and it ran for many years. However, like most games, it did not emotionally challenge the audience. There was no cost for having the 50 50 intuition. In fact, having the wrong intuition made it more surprising and entertaining. However, a slight change to the format can trigger very different emotions. Suppose you and a friend earned your living by playing the game for a year. Instead of a car, you could win $1,000 a day if you picked the winning door. You decided to stick each time and your friend to swap. After a week, you both had earned about the same. However, at the end of the first month, it was clear your friend had won more money. You put this down to luck and confidently continued to apply your stick strategy. Despite this initial confidence, things felt very different at the end of the year. You had expected to earn about the same as your friend. But the final total showed your friend won twice as much money as you. At this point, you would think something was wrong and feel puzzled and stressed. The year-long experience would have eaten away your confidence. Perhaps you would have even been tempted to change strategies.
We can all experience the Monty Hall Problem intuition. Just follow Marilyn's invitation to play the game. All you need are three paper cups, a coin, and some volunteers. Do you or your friends have the Monty Hall intuition? Are you convinced that switching is the best strategy?