The Decision Sciences Journal of Innovative Education

Using Active Learning to Transform the Monte Hall Problem Into an Invaluable Classroom Exercise

Michael Umble

Department of Management, Hankamer School of Business, Baylor University, Waco, TX, email: mike_umble@baylor.edu

Elisabeth J. Umble

Department of Management, Hankamer School of Business, Baylor University, Waco, TX, email: liz_umble@baylor.edu

The Monte Hall Problem refers to the old “Let’s Make a Deal” television game show hosted by Monte Hall, where a contestant is asked to choose between three doors on the stage, winning whatever is behind the door they select. Behind one door is a grand prize (e.g., a car) and behind each of the other two doors is a consolation prize (e.g., a goat). Once the contestant chooses a door, the game show host opens one of the two doors not chosen by the contestant, revealing a goat. Then the contestant is given the option to stay with their original choice or switch to the remaining unopened door. The question is whether the contestant should stay or switch.

Ever since Dr. Andrew Vazsonyi (1999) wrote his article titled “Which Door Has the Cadillac?,” some of us have used the “Monte Hall problem” as an interesting introduction to decision trees. However, as Dr. Vazsonyi indicated, it is difficult to effectively counter the incorrect “common sense” answer that “it should make no difference.” As a result, some students were never fully convinced of the validity of the decision tree solution (the contestant doubles their chances of winning the car by switching). Now this really presents a problem – not only might some students conclude that the decision tree technique is somehow flawed, they might also convince themselves that the instructor is clueless (some don’t need much convincing).

After reading Dr. Vazsonyi’s (2003) recent article “Which Door Has The Cadillac?: Part II,” we concluded that many of our colleagues had probably experienced similar frustrations with the Monte Hall problem.  This article describes how we successfully transformed the Monte Hall problem from a somewhat frustrating experience to an exciting active learning classroom experience for our students.

A PROVEN APPROACH TO OVERCOME RESISTANCE

The source of frustration with the Monte Hall problem is embedded in the fact that it is always difficult to try to counter intuitive beliefs with logic. No matter how airtight the logical or mathematical argument might be, some human beings will simply not easily dismiss internalized beliefs – even if the internalizing took place only moments before. Instead the students must convince themselves that their initial intuitive solution is incorrect! Fortunately, since intuition is partly a function of accumulated past experiences, all that need take place is a new experience that leads each student to conclude, on their own, that their initial intuitive impulse was incorrect.

We have developed an active learning classroom exercise that successfully uses the Monte Hall problem to introduce decision trees. Our approach includes the following sequence of steps.

1.      Describe the Monte Hall problem and ask the students whether they would stay with the original choice or switch. Usually, the vast majority indicate they would stay. Most say it doesn’t matter whether they stay or switch because the car is equally likely to be behind either of the two remaining doors. Do not in any way indicate that they may be incorrect.

2.      Indicate that it might be interesting to simulate the game show. Proceed to demonstrate (see “Simulating Monte Hall Problem” section) how the simulation will be conducted using teams of three students. (As they conduct the simulation, most students quickly conclude that the best strategy is to switch, overcoming their previous incorrect intuition all by themselves.)

3.      Each team records their results on the board in a two by two matrix that includes the stay and switch strategies on one side and the car and goat outcomes on the other side. Then using the recorded simulation results, calculate the relative frequencies of winning the car for each of the strategies of “stay,” and “switch.” If the sample size is sufficiently large, the relative frequencies should closely approximate .33 and .67, respectively.

4.      To help the class internalize that switching is the best strategy, ask for a volunteer to explain why the simulation yielded the observed results. Our experience indicates that the volunteer’s new intuition and explanation will be right on target. Then illustrate what the student has explained (see “Figure 1” and “A Common Sense Explanation” section) and ask if this is a correct summary of their explanation. Wait for their agreement and get a class consensus before proceeding.

5.      Finally, explain how one may not always have the correct insight to resolve complex or non-intuitive problems. And it may not be feasible or easy to gain the experience necessary to obtain the required intuition. Thus, we need straightforward systematic ways of modeling problems to enable us to make the best decisions possible using all the available information. At this point, the students are fully open to learning how to draw and utilize decision trees.

SIMULATING THE MONTE HALL PROBLEM

To simulate the Monte Hall problem, use ordinary playing cards to represent the doors hiding the car and the goats. For example, we use face cards (jack, queen, king, and ace) to represent the door with the car and small cards (two through nine) to represent a door hiding a goat. Demonstrate how the game is played by selecting a face card and two small cards, show them to the class, and explain what they represent. Then pick a student to play the role of the contestant while you play the role of the host. Emphasize that the host should always shuffle the three cards so that the contestant can not see the shuffle. Lay the three cards face down, making a mental note which of the three cards represents the car. Then ask the contestant to choose a card. Of course, one or both of the cards not chosen will be a small card representing a goat. Next, smile as you turn over one of the two cards not chosen, revealing a goat, and say something like “aren’t you glad you didn’t select this one? – you would have won a goat! Now, would you like to stay with your original choice or would you like to switch?” After the contestant answers, then turn over the cards, showing whether they have won a car or goat.

Ask the students to form groups of three to play the game themselves (using teams of two also works, but three seems to be more fun). Give the instructions that one student will play the role of host, one the role of contestant, and one will record the results of the simulations. To enhance the fun and learning, instruct them to switch roles as they perform the simulations.

Our class size is typically about 30 to 40 students, resulting in at least ten teams performing the simulations. We normally ask each team to perform the simulation 20 times. This gives a total of at least 200 observations, which is generally sufficient to derive reasonably accurate relative frequencies of the probabilities of winning the car using the stay and switch strategies.

A COMMON SENSE EXPLANATION

The typical verbal explanation of the solution provided by a student volunteer can be illustrated with the diagram shown in Figure 1. Suppose, for example, that door 1 is selected. There is a 1/3 probability that the car is behind that door. That then requires a 2/3 probability that the car is behind either doors 2 or 3.  When the host then opens either door 2 or door 3 revealing a goat, we then know that the 2/3 probability belongs only to the remaining door, either door 3 or door 2. (Because the car and the remaining goat stay where they were initially placed.) Therefore, if the contestant stays with door 1, they must still have a 1/3 chance of winning the car. If they switch to either door 2 or 3, their chance of winning the car is 2/3.

We also intentionally cover the concepts of expected value of perfect information and expected value of additional information prior to introducing the Monte Hall problem. Then, as part of our explanation, we emphasize that additional information is obtained when the host opens one door containing a goat. And by properly utilizing this additional information, we can improve our chances of winning the car from 1/3 to 2/3. (If we assign a dollar value to the car and the goats, then we can also calculate the expected value of the additional information provided by the host when one of the doors is opened.)

Figure 1.:  Illustration of three doors, hiding one car and two goats.

TESTING THE EFFECTIVENESS OF THE PROPOSED APPROACH

To validate the effectiveness of the approach proposed in this paper, we conducted an experiment in four sections of a required operations management class. Two instructors each presented the Monte Hall problem/decision tree solution to two class sections. Each instructor conducted the exercise for one section (control group) without using the proposed approach – that is presenting the Monte Hall problem and then developing the solution using a standard decision tree analysis without benefit of the simulation exercise and approach proposed in this paper. Each instructor then conducted the Monte Hall problem in a second class (experimental group) following the approach proposed in this paper. At the conclusion of each exercise, a questionnaire utilizing a seven point Likert scale was distributed to the students. The questionnaire included the following four questions:

1. This exercise was fun.

2. This exercise held my interest.

3. I am convinced that the probability of winning the car is 2/3 if you switch and 1/3 if you stay.

4. Decision trees seem to be useful for structuring and evaluating certain types of problems.

The seven Likert scale response choices included: strongly disagree, disagree, slightly disagree, neither agree nor disagree, slightly agree, agree, strongly agree. The responses were recorded and scored using a -3, -2, -1, 0, 1, 2, 3 numerical scale. The average numerical score for each of the four questions for instructor 1’s control and experimental groups and instructor 2’s control and experimental groups are presented in Figure 2.

Figure 2.: Survey results from the Monte Hall problem

Figure 2 indicates that students in the experimental groups (following the proposed approach) responded much more favorably to each of the four questions than students in the control groups. In addition, a chi-square test was conducted to test for a significant difference in responses  between the control and experimental groups for each instructor for each of the four questions.  Each of the eight tests were statistically significant at the .01 level. 

Thus, the evidence indicates that the approach proposed in this paper promises to transform the Monte Hall exercise into an exciting active learning classroom experience that students find to be fun and interesting. The proposed approach also appears to provide a strong intuitive understanding of the 2/3 - 1/3 probability solution and sets the stage for a powerful and convincing introduction as to the usefulness of decision trees for analyzing certain types of decision problems.

CONCLUSION

We do not present the decision tree solution to the Monte Hall problem here since it is clearly explained in Dr. Vazsonyi’s second article and is well known. Moreover, the purpose of this article was to present an enjoyable active learning classroom exercise that has been classroom tested and proven in multiple sections of operations management classes. Instructors who have used the approach proposed in this paper unanimously concur that the previously encountered skepticism and resistance is completely eliminated, making it easy to achieve the desired learning objectives.

REFERENCES

Vazsonyi, A. (1999). Which Door Has the Cadillac? Decision Line, 30(1), 17-19.

Vazsonyi, A. (2003). Which Door Has the Cadillac?: Part II. Decision Line, 34(3), 15-17.