Landon Baker 12/6/12 Essay #3 Math 89S GTD Exploring the Monty Hall Problem Problem solving is a human endeavor that evolves over time. Children make lots of mistakes, primarily because they have fewer experiences to draw from and therefore have limited information to eliminate poor choices and select good ones. As time goes by and their experiences grow in number and frequency, a natural filtering process occurs which allows the human brain to create logical pathways, thereby developing something we call intuition. We come to rely on our intuition as we get older, sometimes to our benefit and other times to our detriment. Previously held truths might be questioned or made ambiguous by certain scientific or mathematical evidence, and prove to be counterintuitive and, therefore, uncomfortable. For example, long ago beliefs regarding orbital pathways in the solar system and the shape of the Earth were ultimately disproved despite people s intuitive trust in the theories that existed earlier in history. We may now know for certain that the Earth is round, but we can all appreciate why so many people would devotedly believe that such a concept was intuitively impossible. Game theorists attempt to utilize knowledge of intuitive pathways in order to come up with predictive models for human behavior, often proving their theories mathematically. Sometimes, they uncover a game that plays on people s intuition, but inadvertently prove that behavior isn t always as rational as the mathematics would suggest. The Monty Hall problem (or Paradox) is based on an old television game show called Let s Make a Deal, and was named for the show s famous host, Monty Hall. It is a highly counter-intuitive statistics puzzle that confounds most people to this day,
despite a multitude of mathematical proofs and computer-based simulations demonstrating the correct response of the player. It is fascinating that the solution to the Monty Hall problem leaves people mind-boggled, but the explanation lies in the fact that people feel compelled to trust their own intuition rather than accept the mathematics behind the solution. The Monty Hall problem was presented as follows by Marilyn vos Savant in her September 1990 column in Parade Magazine: 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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? 1 For clarification, the host is constrained to open one of the unchosen doors that contains a goat and to always make the offer to the contestant to switch. With only two doors left to choose from after Monty reveals one of the losing doors, most players assume that their odds of choosing the door that hides the car have just increased from 1/3 to 1/2 when, in fact, their odds have just increased to 2/3 if they simply switch to the unchosen door. Many studies have analyzed the behavior of the contestants when confronted with this decision, and the results show that most people ultimately decide to stay with their original choice. One study in particular tested 228 subjects, and only 13 percent of these contestants decided to switch doors. 2 The main reason for the huge disparity between what people do and what people should do is based predominately on intuition. While faced with the problem, nearly all people believe that each of the two unopened doors has an equal chance of containing the 1 vos Savant, Marilyn (1990a). "Ask Marilyn" column, Parade Magazine p. 16 (9 September 1990). 2 Granberg, Donald and Brown, Thad A. (1995). "The Monty Hall Dilemma," Personality and Social Psychology Bulletin 21(7): 711-729.
car and, therefore, conclude that switching doesn t matter. Intuitively this makes sense because if one of the doors were opened, one would naturally think that each of the remaining doors has a 50 percent chance of containing the car, since each door either contains a car or a goat. However, relying on intuition to solve the Monty Hall problem has been proven not to be the best approach. Additionally, there is a psychological explanation for the overwhelming majority of contestants choosing to stay with their original door that is based on both the endowment effect, which says that people have a tendency to overvalue what they already own, 3 and the status quo bias, which states that people have a penchant for the current state of affairs or the status quo. 4 These two cognitive explanations definitely contribute to the way contestants act during the game, as the status quo bias actually had a significant impact on the results of my own survey. I conducted a survey of 20 Duke students in which I created a very similar scenario to the Monty Hall problem and asked what each would do if presented with that situation. The scenario read: 3 Kahneman, D., J.L. Knetsch and R.H. Thaler, 1991. Anomalies: The endowment effect, loss aversion, and status quo bias. The Journal of Economic Perspectives. 4 Samuelson, W. and R. Zeckhauser, 1988. Status quo bias in decision making. Journal of Risk and Uncertainty, 1, pp. 7 59.
There are three hats. There is one million dollars under one hat and nothing under the other two. You choose a hat but, before you turn over the hat, your friend turns over another hat that has nothing under it. Your friend then asks you if you want to keep the hat you originally chose or if you want to switch and choose the other remaining hat. What would you do? (A) Stay (B) Switch Does it matter? (A) Yes (B) No Additionally, I asked each participant to write down his/her gender and what school he/she attends (the Trinity School of Arts and Sciences or the Pratt School of Engineering). Following the survey, I asked each student if they knew what the Monty Hall problem was, as I only analyzed the results of those who were not familiar with the paradox. The three choices were (1) Stay, (2) Switch, and (3) It doesn t matter. The results of the survey were very lopsided, as the vast majority of students chose stay for the first question and no for the second question. Although seven students chose yes for the follow-up question, all seven acknowledged that they answered that way simply because they thought it was a trick question, and not because they had figured out the solution mathematically. The principle reason for the overwhelming majority of participants choosing to stay with their original choice appears to be the status quo bias. Following most of the surveys, I asked the students what caused them to answer the way that they did and, overwhelmingly, the most popular response was that they felt more comfortable and confident sticking with their original gut feeling. This reasoning is the basis of the status quo bias, arguing that people fear the
possibility of getting the answer wrong after doubting their original choice and choosing an alternative solution. Certain students were eliminated as candidates from the survey because they acknowledged that they were familiar with the Monty Hall game. Interestingly, all those students were either in Pratt or were male. While the sample size of this survey is too small to be statistically significant, I became curious about whether or not there would be a general discrepancy between the answers given by males and females, and the answers given by students in Trinity and Pratt. I hypothesized that both more male students and more students in Pratt would choose to switch hats (and thus answer correctly) largely due to the old stereotype that males tend to think more quantitatively than females. In some ways, my hypothesis turned out to be correct: I found it significantly more difficult to find males who were not familiar with the Monty Hall problem than it was to find females with no prior knowledge. Additionally, the majority of students who were familiar with the problem were also in Pratt, which is also not surprising given that approximately 70 percent of the students in Pratt are, indeed, male. However, of those students questioned who did not know about the Monty Hall problem prior to the survey, there was no discernable trend or pattern between the answers of males and females, leading to the conclusion that no relationship exists between the success rate of males and females who are not already familiar with the problem. Therefore, the results of my survey also show that, without any prior knowledge, males and females have roughly the same level of intuition. Yale psychologists performed an interesting study on monkeys in order to determine if their preferences and choice rationalization showed a significant correlation
with human behavior during a simulated variation of the Monty Hall problem. 5 First, the conductors of this experiment sought out different colors of M&Ms and identified three colors that were equally preferred by a monkey: red, blue, and green (as an example). Then, the researchers gave the monkey a choice between two of them. Next, the monkey was given the choice between the loser of the first matchup and the M&M not used in the first matchup. What they found was that 2/3 of the time the monkey rejected the M&M that was originally rejected in the first matchup. This corresponds with the theory of choice rationalization, which states that once we reject something, we are inclined to think that we never liked it anyway. This is another psychological explanation for people s behaviors while confronted with the Monty Hall problem: since the contestant already chose not to pick the other unopened door, he/she will reject it again because they decided not to choose it originally. Although the study described above was performed on monkeys, their behavior during the experiment parallels the behavior of those answering the Monty Hall problem. Up to this point, I have only revealed the correct solution to the Monty Hall problem (the player should switch doors as his/her odds of choosing the car increase to 2/3 if he/she does so) and discussed what accounts for people s proclivity to choose the incorrect solution. However, understanding the solution to the problem is essential in figuring out where the contestants go wrong when they rely on intuition to solve the problem. A problem that was first raised by in a 1975 letter by Steve Selvin 6 to the American Statistician, and subsequently addressed by Marilyn vos Savant in 1990, left 5 Tierney, John. "And Behind Door No. 1, a Fatal Flaw." The New York Times. The New York Times, 08 Apr. 2008. Web. 05 Dec. 2012. 6 Selvin, Steve (February 1975), "A problem in probability (letter to the editor)", American Statistician 29 (1): 67
people so bewildered that they began finding differing ways to solve the problem and arrive at the proper, counter-intuitive answer. Each of these proofs includes a certain combination of intuitive thinking and problem solving. In the following section, I will cover solutions that combine those two skills in different proportions by discussing vos Savant s method, Carlton s method, and Bayes theorem. Marilyn vos Savant is listed in the 1985 Guiness Book of World Records as having an IQ of 190. Also a writer, Marilyn vos Savant is known for a column that appeared in Parade magazine on September 9, 1990. In this column, she presented the Monty Hall problem and then proceeded to answer the question by stating that a switch guaranteed a 2/3 success rate of choosing the car. Although her explanation was rather mathematically involved, her solution was based off the following table that represents all of the different outcomes of the problem when originally choosing door 1: Behind Door 1 Behind Door 2 Behind Door 3 Result if Staying with Door 1 Result if Switching Doors Car Goat Goat Car Goat Goat Car Goat Goat Car Goat Goat Car Goat Car The first three columns of the table lay out all of the possible combinations of cars and goats behind the three doors. The three possible orderings are: car, goat, goat; goat, car, goat; or goat, goat, car. Consequently, if the contestant chooses door 1 and decides to stay with his/her original choice of door, the fourth column shows what the contestant would win for all three possible combinations of cars and goats. Similarly, the fifth column contains the possible outcomes when switching doors. Evidently, one has a 2/3
probability of winning the car when switching and a 1/3 chance when staying with one s original choice. This table is independent of which door the contestant originally chooses, so the problem would yield identical solutions if the contestant were to originally choose door 2 or door 3. Matthew Carlton published perhaps the most intuitive explanation of the solution to the Monty Hall problem in 2005. 7 He stated that a player s strategy to switch only loses if he/she initially chose the car. As a result, since the original probability of picking a car is 1/3, switching must win 2/3 of the time. In other words, if the contestant picks a goat, he/she will always win by switching, as the other door containing a goat can t be picked since the game show host already opened it. Additionally, if the contestant originally picks the car, he/she loses by switching since the other two doors contain goats. Consequently, the contestant wins by switching if he/she originally picks a goat and loses by switching if he/she originally picks the car. Since there is a 2/3 chance of originally picking a goat, the contestant has a 2/3 chance of winning by switching. Another way of looking at this problem from an intuitive standpoint is to imagine that there are 1,000,000 doors rather than three doors. If the contestant chooses one door, that leaves 999,999 doors left to open. If the game show host opens 999,998 doors and reveals goats behind each one, is it wise to stay with the original door chosen (with initial odds of 1 in a million) or switch to the only other door remaining? Using larger numbers, it is more obvious to see that with every turn of a door, the game show host is revealing more and more information to the contestant, which should make it obvious that the host 7 Carlton, Matthew (2005). "Pedigrees, Prizes, and Prisoners: The Misuse of Conditional Probability". Journal of Statistics Education [online] 13 (2). Retrieved 2010-05-29.
may know something that the contestant does not. A switch of doors seems completely rational given this new scenario. The final way to solve the Monty Hall Problem that I will discuss, Bayes Theorem, is the most mathematically involved proof of the solution. First off, Bayes Theorem states: P A B = P B A P A P B Literally speaking, this equation means that the probability of A, given B, is equal to the probability of B, given A, times the probability of A, all divided by the probability of B. However, the Monty Hall problem requires us to insert two pieces of information as opposed to the one piece shown in the above equation. As a result, the equation used to solve the Monty Hall problem is the following: P(A B, C) = P(C B, A) P(A B) P(C B) In this equation, we will designate the following definition to A, B, and C: Let A = the unopened door (door 3) contains the car Let B = originally choosing door 1 Let C = the host opening door 2 Consequently, the equation states that the probability of the car being in door 3, given the fact that the contestant originally choses door 1 and the host opens door 2, is equal to the probability that the host opens door 2, given the fact that the contestant chooses door 1 and door 3 contains the car, times the probability of door 3 containing the car, given the fact that the contestant originally chooses door 1, all divided by the probability of the host opening door 2, given the fact that the contestant chooses door 1.
It s easier to solve this equation by dividing the equation into its various components. First off, P(C B, A) = 1 because the host never selects the door you choose or the one with the car. Secondly, P(A B) = 1/3 because the position of the car had an initial probability of 1/3, and this doesn t change based on the choice. Finally, P(C B) = 1/2 because the host can either open door 2 or door 3 after the contestant chooses door 1, so there is a 1/2 probability of the host opening door 2. If we plug these three values into the equation, we get: P(A B, C) = 1 1/3 1/2 = 2/3 This answer means that there exists a 2/3 probability that the unopened door contains the car, and therefore, the contestant will win the car 2/3 of the time if he/she decides to switch. A similar equation is used to figure out the probability that the contestant s original choice contains the car. Evidently, however, the answer is 1/3 since the car must be in either the originally chosen door or the unopened door. 8 The Monty Hall problem is a fascinating representation of the failure of intuition to guide an individual to the most efficient and rewarding outcome. Although Carlton s method for solving the problem is intuitively based, one must still take some quantitative steps in order to complete the proof. In fact, multiple studies, including my survey, show that contestants relying solely on intuition end up staying with their original choice due to psychological factors such as the status quo bias and the endowment effect. Despite the multiple proofs of the problem, there are still many people who doubt these solutions and insist that each of the two remaining doors has a 1/2 chance of containing the car. These 8 "Bayes Theorem and the Monty Hall Problem." Formalised Thinking. N.p., 10 Oct. 2010. Web. 05 Dec. 2012.
individuals are somehow choosing to ignore the importance of the information that the game show host divulged to them by opening a losing door after the fact. Instead of synthesizing this new information, and realizing that the odds of the game have changed, these players break the natural learning process that comes from new experiences. Perhaps the explanation lies in the fact that, when presented with new information, people are hesitant to override their well-developed sense of intuition and, consequently, risk falling into the same trap as civilizations of long ago that refused to believe that the Earth was anything but flat.
Works Cited vos Savant, Marilyn (1990a). "Ask Marilyn" column, Parade Magazine p. 16 (9 September 1990). Granberg, Donald and Brown, Thad A. (1995). "The Monty Hall Dilemma," Personality and Social Psychology Bulletin 21(7): 711-729. Kahneman, D., J.L. Knetsch and R.H. Thaler, 1991. Anomalies: The endowment effect, loss aversion, and status quo bias. The Journal of Economic Perspectives. Samuelson, W. and R. Zeckhauser, 1988. Status quo bias in decision making. Journal of Risk and Uncertainty, 1, pp. 7 59. Tierney, John. "And Behind Door No. 1, a Fatal Flaw." The New York Times. The New York Times, 08 Apr. 2008. Web. 05 Dec. 2012. Selvin, Steve (February 1975), "A problem in probability (letter to the editor)", American Statistician 29 (1): 67 Carlton, Matthew (2005). "Pedigrees, Prizes, and Prisoners: The Misuse of Conditional Probability". Journal of Statistics Education [online] 13 (2). Retrieved 2010-05-29. "Bayes Theorem and the Monty Hall Problem." Formalised Thinking. N.p., 10 Oct. 2010. Web. 05 Dec. 2012.