MSTE
Office for Mathematics, Science, and Technology Education
College of Education @ University of Illinois

# Monty's Dilemma

## The Concept:

In a certain game show, a contestant is presented with three doors. Behind one of the doors is an expensive prize, behind the others is nothing. The contestant is asked to choose a door. The game show host, Monty, then opens one of the other doors to reveal a that there is nothing behind it. The contestant is then asked if she/he would like to stick with the original door or switch to the remaining door.

## Instructions:

Choose a door by clicking on it. Monty opens a door with nothing behind it, then you are presented with a choice to either stick with the door you originally picked or to switch to the other unopened door.

User the controls on the right side to run a number of situations without having to go through the process of picking a door and choosing to switch or to stay. Choose the number of trials to run and what strategy to use, then click the Run Trials button. The results are displayed in the table on the lower right.

### Controls:

• Doors: Click on a door to choose it.
• New: Start a new simulation, ignoring the current experiment.
• Stay, Switch, Flip:
• Stay: Stay with the door you chose
• Switch: Change to the other unopened door.
• Flip: Flip a coin. If heads come up, you will stay with the door you chose. If you get tails, you switch to the other door.
• Run Trials: Run simulated trials. You specify the number of trials to run: 10, 100, or 1000. You choose the strategy to use for every run: Stay, Switch, or Flip.

## The Purpose:

What are the chances of winning if you always choose to Stay? Switch? The results might surprise you. Attempt to understand the underlying notions of statistics and probability through various runs. The results, the number of wins, for each strategy is shown in the table in the lower right corner. Use this information to come up with an equation for the chances of winning.

## Example Run:

The contestant chooses door 2. Monty opens door 3 to reveal that there is nothing behind it. Monty knows which door has the prize and always opens one that has no prize behind it. Should the contestant

1. Stick with Door 2?
2. Switch to Door 1?
3. Does it matter, i.e., could you flip a coin to decide?

## Monty's Dilemma Extended:

Still not sure why the results are not what you expected? We've extended the idea of Monty's Applet to include more doors than 3. The concept is still the same: Monty opens all the doors except the one you chose and the one that contains the prize. Now you can try this same game with many more doors. The simulation with more doors should make it even simpler to see how and why the probabilities for success have these specific values.

## References:

The original Monty's Dilemma Applet was the tool used for a study by Jesse "Jay" Wilkins and George Reese.

The pedagogical persuasiveness of simulation in situations of uncertainty, Proceedings of the Twentieth Annual Meeting North American Chapter of the International Group for the Psychology of Mathematics Education, November 1998. Wilkins, J. L. M. & Reese, G. C. (1998). Vol. 1, (p. 411).

In the study we examined the effectiveness of this computer simulation in changing students' decision-making in conditions of uncertainty. For a copy of the paper contact Jay Wilkins at Virginia Polytechnic Institute and State University.

The Monty's Dilemma problem has been the subject of many articles. A few choice examples include:

• Engel, E., & Venetoulias, A. (1991). Monty Hall's probability puzzle. Chance, 4 (Spring), 6-9.
• Morgan, J. P., Chaganty, N. R., Dahiya, R. C., & Doviak, M. J. (1991). Let's make a deal: The player's dilemma. The American Statistician, 45(4), 284-287.
• Shaughnessy, M. J., & Dick, T. (1991). Monty's dilemma: Should you stick or switch? Mathematics Teacher, 85(April), 252-256.

The probability underlying the problem is discussed in the following texts:

• Konold, C. (1994). Teaching probability through modeling real-world problems. Mathematics Teacher, 87(4), 233-235.
• Shaughnessy, M. J. (1991). Misconceptions of probability: From systematic errors to systematic experiments and decisions. In A. P. Shulte & J. R. Smart (Eds.), Teaching statistics and probability, 1981 yearbook of the National Council of Teachers of Mathematics (pp. 90-100). Reston, VA: NCTM.
• Travers, K. J. (1981). Using Monte Carlo methods to teach probability and statistics. In A. P. Shulte (Ed.), Teaching statistics and probability (Vol. 1981 Yearbook, pp. 210-219). Reston, Virginia: The National Council of Teachers of Mathematics.
• Watkins, A. E. (1981). Monte Carlo simulation: Probability the easy way. In A. P. Shulte (Ed.), Teaching Statistics and Probability (pp. 203-209). Reston, Virginia: The National Council of Teachers of Mathematics.

## The Credits:

Applet designed and coded by Pavel Safronov. Based on applet and activity by George Reese.

Download the source code in a txt file. To run the source code, change the extension to .java