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 counter-

intuitive 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 Selvin6 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:

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

Behind Door 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

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  ×  1/31/2 = 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.