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THE MONTY HALL PROBLEM

Throughout the many years of Let's Make A Deal's popularity, mathematicians have been fascinated with the possibilities presented by the "Three Doors" ...  and a mathematical urban legend has developed surrounding "The Monty Hall Problem."  The CBS drama series NUMB3RS featured the Monty Hall Problem in the final episode of its 2004-2005 season.  The show's mathematician offered his own, very definite solution to the problem involving hidden cars and goats.

The 2008 movie 21 opens with an M.I.T. math professor (played by Kevin Spacey) using the Monty Hall Problem to explain mathematical theories to his students.  His lecture also includes the popular "goats and cars behind three doors" example favored by many versions of the Problem. 

The London FINANCIAL TIMES published a column about the Monty Hall Problem on August 16, 2005, declaring positively that "the answer is, indeed, yes: you should change."  However, the columnist, John Kay, notes that "Paul Erdos, the great mathematician, reputedly died still musing on the Monty Hall problem."  The column resulted in several  letters published on the "Leaders and Letters" page of the FINANCIAL TIMES on August 18 and 22 - and two follow-up columns by Mr. Kay on August 23 (So you think you know the odds) and August 31 (The Monty Hall problem - a summing up) in which he acknowledges that he received "a large correspondence on Monty Hall."

 

The controversial "problem" began when Marilyn vos Savant published a puzzle in her Parade Magazine column.  One of her readers posed the following question:

 “Suppose you’re on a game show, and you’re given a choice of three doors: Behind one door is a car; behind the others, goats.  You pick a door, say number 1, and the host, who knows what’s behind the doors, opens another door, say number. 3, which has a goat.  He says to you, ‘Do you want to pick door number 2?’  Is it to your advantage to switch your choice of doors?” 

 

Ms. Savant, who’s listed in the Guinness Book of World Records Hall of Fame for “Highest IQ” (228), answered “Yes.”  Because of the estimated 10,000 letters she received in response, she published a second article on the subject.  A scholarly analysis of the letters she received was compiled by Donald Granberg, Ph.D., in a paper entitled "The Monty Hall Dilemma: To Switch or Not to Switch."  

 

Ms. Savant's book The Power of Logical Thinking is devoted to the dilemma, providing a detailed account of the history of the Monty Hall problem. In addition to the rigorous account of the mathematical assumptions that lead to Marilyn's solution as the correct one, the book includes Granberg's paper as an appendix. 

 

Due to the fervor created by Ms. Savant’s two columns, the New York Times published a large front page article in a 1991 Sunday issue which declared:  “Her answer... has been debated in the halls of the C.I.A. and the barracks of fighter pilots in the Persian Gulf.  It has been analyzed by mathematicians at M.I.T. and computer programmers at Los Alamos National Laboratory in New Mexico.  It has been tested in classes ranging from second grade to graduate level at more than 1,000 schools across the country.” 

 

Mark Haddon's 2003 best-selling novel the curious incident of the dog in the night-time includes a section in which the main character, a mathematically gifted and socially challenged teenager,  discusses the Monty Hall Problem at length and presents the following equation to illustrate his conclusions:

Let the doors be called X, Y and Z.

Let Cx be the event that the car is behind door X and so on.

Let Hx be the event that the host opens door X and so on.

Supposing that you choose door X, the possibility that you win a car if you then switch your choice is given by the following formula

P(Hz ^ Cy) + P(Hy ^ Cz)

= P(Cy) P(Hz Cy) + P(Cz) P(Hy Cz)

= (1/3 1) + (1/3 1) = 2/3

 

 

In 2004, Cornelsen, a German textbook company, published a mathematics text entitled  Mathematik Stochastik: Grund- und Leistungskurs / Orientierungswissen Analytische Geometrie in which the first chapter is devoted to a discussion of the Monty Hall Problem - complete with diagrams of the goats and cars - and photos of Marilyn vos Savant and the Three Doors.

 

 

 

In November, 2004, Ivan Moscovich published a book of mathematical puzzles entitled The Monty Hall Problem & Other Puzzles.

 

 

 

 

The Monty Hall Problem is addressed in several scholarly books on probability and problem solving -- including David Sirzaker's Elementary Probability (2003)  - Decisions, Decisions: The Art of Effective Decision Making (2002) by David A. Welch -  Steven Krantz's Techniques of Problem Solving (1996) - and ...

 

 

 

Struck by Lightning: The Curious World of Probabilities (2006) by Jeffrey Rosenthal.

 

Rosenthal concludes that the answer to the problem is counter-intuitive:  You improve your chances of winning by twofold if you change your choice after one door has been revealed.  

 

 

 

 

 

The Nature of Mathematics

There's a discussion of the Monty Hall Problem in the math textbook The Nature of Mathematics  written by Karl J. Smith and published by Thomson Learning (2006).

 

 

 

In The Drunkard’s Walk (2008) Leonard Mlodinow provides readers with a guide to how the mathematical laws of randomness affect our lives.  He uses the Monty Hall Problem as an example of the "law of sample space."  Mlodinow notes that The Monty Hall problem "has immortalized both Marilyn (vos Savant) and Let's Make a Deal ... It had to come as a surprise to the show's creators that after airing 4,500 episodes in nearly twenty-seven years, it was this question of mathematical probability that would be their principal legacy.".

 

 

 

 

 

Links to websites devoted to 

"The Monty Hall Problem"

Door 1Door 1Door 3

*art work courtesy of cartalk.cars.com

 

 

 Monty Hall Problem Stay or switch?? Explanation of the "problem" with a cleverly animated simulation of the game - with goats and cars behind the doors.
 Problem Explained A scholarly explanation of the controversy, with a really old photo of Monty Hall.
Three Doors Simulation A Monty Hall Problem Simulator by David Welch.
 Monty's Dilemma An explanation of the "Three Door Problem" - and a java applet to play the game.
 Pick A Door Play the game and pick a door to get a car or a goat!
 Donkey Deal This applet lets you play the game for donkeys or cash!
 Pick A Card An explanation of the Problem using playing cards - and source code for a C program to simulate the problem.

As anyone who's surfed the web knows, there are hundreds of sites addressing this "problem" - and numerous scholarly solutions.

Here's a response from Monty to a query about the "Monty Hall Problem":

May 12, 1975

Mr. Steve Selvin
Asst. Professor of Biostatistics
University of California, Berkeley

Dear Steve:

Thank you for sending me the problem from "The American Statistician."

Although I am not a student of a statistics problems, I do know that these figures can always be used to one's advantage, if I wished to manipulate same. The big hole in your argument of problems is that once the first box is seen to be empty, the contestant cannot exchange his box. So the problems still remain the same, don't they. . . one out of three. Oh, and incidentally, after one is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so. It was always two to one against him. And if you ever get on my show, the rules hold fast for you -- no trading boxes after the selection.

Next time let's play on my home grounds. I graduated in chemistry and zoology. You want to know your chances of surviving with our polluted air and water?

Sincerely,
Monty

In The American Statistician, August 1975, Vol. 29, No. 3, Steve Selvin quoted Monty's letter and responded:  "I could not have said it better myself."

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