From the
Exploring Data website - http://curriculum.qed.qld.gov.au/kla/eda/
© Education Queensland, 1997
The Monty Hall Puzzle
The Monty Hall puzzle swept through high school maths departments in Rockhampton in the mid-1990s. Even though the solution needs no more than basic probability concepts many maths teachers initially arrived at the wrong answer, and some needed a fair bit of convincing that their answer was wrong.
The Puzzle
A television game show called The Price is Right compered by Monty Hall was very popular in the U.S. in the 1950s and early 1960s. I used to watch it occasionally as it would often be on when I came home from school.
The puzzle assumes the following scenario, which is similar to what actually happened: There is a single winner for each show. In the last few minutes the winner had the opportunity to win a valuable prize. The contestant was given the choice of three doors - one of which concealed the valuable prize with the others concealing prizes of relatively little value. After the contestant made a choice Monty Hall would open one of the other doors and reveal the prize behind that door. It was always a prize of little value. He then gave the contestant the option of staying with the initial choice of door or switching to the other door. The door that the contestant finally chose was opened to reveal the prize.
The puzzle: is there any benefit in switching, or is there any benefit in not switching, or does it make no difference what the contestant does?
If you have never seen this puzzle before, I suggest you try to solve it before looking at the solution.
Pedagogy
Once someone has solved the puzzle it is easy for them to forget that the answer is counter-intuitive to many people. I think the best use of this puzzle in the classroom would be to present it as a challenge to students working in small groups, and warn them that the puzzle is trickier than it first appears and that many mathematicians initially were unable to solve the problem correctly. Challenge them to find the solution and then write up the solution in such a way that a fellow student would understand why their answer is correct. It is not an easy task - whenever this puzzle crops up on a maths education mailing list there are a flood of emails trying to convince the doubters - and for the most part these doubters are trained maths teachers.
Some teachers introduce this puzzle as a challenge at the beginning of a probability unit, and leave it to the students to find their own method of solution. Others use it as an application of conditional probability. Given its difficulty it certainly wouldn't be fair to use on a supervised assessment piece, and it is too well-known to use in an unsupervised assessment setting.
There are a number of websites devoted to this puzzle. Here are some good ones.
The Let's Make a Deal applet (requires a Java-enabled browser)
This site contains a java applet that allows students to simulate playing the game. It keeps a record of the number of wins if the player stays and if the player switches. Twenty games of each should be enough to convince the student that the theoretical answer is correct.
This page also allows students to conduct repeated trials and see the outcomes from both switching and not switching. It doesn't use Java, so is accessible to more viewers. This page also discusses an extension to the problem, involving four doors.
The WWW Tackles The Monty Hall Problem
If you wish to visit other pages devoted to this puzzle, then this is a good place to start. It contains numerous links to other Monty Hall pages.