Monty Hall

(Authors: Ken McGraw,  Editor:   Jean Mandernach)

Introduction

One of the triumphs of science is its capacity to demonstrate the validity of conclusions that at first seem counterintuitive and clearly at odds with commonsense. Science makes these demonstrations using the empirical method, which uses facts to describe and explain observable pheoneomena, regardless of what people might believe.

The Monty Hall problem uses the empirical method to highlight the difference between formal and informal problem solving. The Monty Hall problem is a game patterned after "Let's Make A Deal," a popular game show hosted by Monty Hall. In this game, there are three doors, of which contestants choose one. Behind one is a new car and behind the other two are goats. Once the contestant chooses a door, Monty Hall opens one of the doors that was not chosen to show the location of one of the goats. Having seen one of the goats, the contestant has the opportunity to stay with the original door selection or to switch to the door that Monty Hall did not open. The question is "Should the contestant switch doors?"

Monte Hall table
Figure 1

Using an informal problem solving strategy, most people conclude that it doesn't matter; that the contestant has an equal probability of winning whether they switch doors or not. The informal logic is that there are two doors remaining, so the contestant has a 50-50 chance of being right whether they stay or switch. But, although this logic seems to make sense, it is not accurate.

In contrast, application of a formal problem solving strategy shows that probability of selecting the correct door changes once the host has revealed one of the goats. In the initial choice, the contestant has a 1-out-of-3 chance of selecting the car (or a 2-out-of-3 chance of selecting a goat). But, once the host has revealed a goat, the probabilities change. Now, the contestant who switches doors has a 2-out-of-3 chance of winning the car. The following example shows the new probability:

Contestants Initial Choice Monty Hall Reveals Results if STAY Results if SWITCH
Door A Door B WIN LOSE
Door B Door C LOSE WIN
Door C Door B LOSE WIN
Figure 2

Regardless of the contestant's initial choice, their odds of winning the car increase from 1-out-of-3 to 2-out-of-3 by choosing to switch doors once Monty Hall has revealed one of the goats. It is important to note that the favorable odds for switching apply only when the host (e.g., Monty Hall) knows where to find the car and the goats. If the host is naïve, the advantage of switching no longer obtains.

One of the great things about the empirical method is that it allows us to discover the truth about things that we don't understand or that are counter-intuitive. In fact, empirically learning the truth of things frequently precedes our understanding of them.

Procedure

In this experiment, participants are game show contestants trying to win a car by selecting it from one of three available doors. Contestants select one of the three doors. Once they make their selection, one of the host opens a door and reveals a goat. Contestants then have the option to "Stay" with the original choice or to "Switch" to the remaining door. The winning door is revealed and a tally reports the percentage of wins based on whether the contestant choose to switch or stay.

Contestants can play as many games as they like using the "Play Again" button that is available at the end of each game. Contestants should continue to play until they have convinced themselves empirically of the correct answer to the question "Are you as likely to win by staying as by switching?" When finished, the contestant selects the "Quit" button.

Design

There are two choices available each time the game is played: stay or switch. Contestants choose one or the other and then observe whether the choice leads to a win or a loss. The results are tallied so that contestants can observe over time whether one choice is as likely as the other to produce a win.

Data Download Format & Data Analyses

The data consist of counts of the number of "Switch" and "Stay" choices and of the number of "Switch" and "Stay" wins. From these, one can compute percentages reflecting the empirical probabilities for winning by dividing the number of wins by the number of games played. For a large enough dataset, the empirical percentages will closely approximate the true probabilities.

Sample data image from the Monte Hall experiment
Figure 3

The first six columns provide demographic and classification data (participant ID number, class ID number, gender, age, study completion date and total time spent in the experiment). The remaining columns indicate the the number of "Switch" and "Stay" choices and of the number of "Switch" and "Stay" wins. The variable codes are as follows:
StayWins - Clicking on Stay and wnning
StayGames - Total number of times clicking on Stay
SwitchWins - Clicking on Switch and winning
SwitchGames - Total number of times clicking on Switch

Applications/Extensions

On learning the outcome of the experiment, which consists of empirical estimates of the true probabilities of winning when electing the "Stay" or "Switch" strategy, it is valuable to discuss other differences between formal and informal problem solving strategies. For example, discussions could examine similar game shows (such as "Deal or No Deal") in which the "switch advantage" is not present. Or, applications could be made to other counter-intuitive problems (such as the "three cards problem" or the "Three Prisoners Problem") that use the same probability logic. In addition, discussions might address formulas for statistical probability, the use of decision trees, Venn diagrams, or Bayes' theorem.

References

Bapeswara-Rao, V. V., & Rao, M. B. (1992). A three-door game show and some of its variants. 
	The Mathematical Scientist, 17, 89-94.
	
Bohl, A. H., Liberatore, M. J., & Nydick, R. L. (1995). A tale of two goats... 
	and a car, or the importance of assumptions in problem solutions.
	Journal of Recreational Mathematics, 27, 1-9. 
	
Tierney, J. (1991, July 21). Behind Monty Hall's doors: Puzzle, debate and answer?, The New York 
	Times, pp. A1.
	
vos Savant, M. (1990, February 17). Ask Marilyn column, Parade Magazine, pp. 12.