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.
What do you think is the best strategy? Should the contestant stick with their original door, switch to the other closed door, or is the probability of winning the same either way?
This is known as the Monty Hall problem, named after the game show host Monty Hall who offered this scenario to contestants on the game show Let's Make a Deal.
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.
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 summary results for each strategy are collected in a table below. Use this information to come up with the probability of winning for each strategy.
To help demonstrate the probabilities more clearly, you can add more doors. After you choose a door, Monty will open all but one of the remaining doors. Now should you stick with your door or switch to the other door? How likely is it that you chose the winning door with your first pick as you increase the number of doors?
| Stick | Switch | Flip | |
|---|---|---|---|
| Wins | 0 | 0 | 0 |
| Losses | 0 | 0 | 0 |
| Total Trials | 0 | 0 | 0 |
| Percent Wins | 0% | 0% | 0% |
Try running 1,000 trials for each strategy: Stick, Switch, and Flip. For 3 doors, you will find that the Switch strategy wins about two-thirds of the time. The Stick strategy wins about one-third of the time. The Flip strategy wins about half the time. Why?
Consider this: the probability that you will pick an incorrect door on your first guess is 2/3. When this happens, Monty is forced to open the one remaining GOAT door. By switching, you will then get the winning door on the 2 out of 3 times that you were incorrect on the first guess. Thus by switching, you turn the odds in your favor. Increasing the number of doors turns the odds even more in your favor if you switch doors.
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:
The probability underlying the problem is discussed in the following texts:
Original applet and activity by George Reese, updated by Pavel Safronov, converted to JavaScript and HTML5 by Michael McKelvey.
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