## 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 C*_{x} be the event that the car is behind
door X and so on.

*Let H*_{x} 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(H_{z} ^ C_{y}) + P(H_{y} ^ C_{z})

= P(C_{y})
P(H_{z}
C_{y}) + P(C_{z})
P(H_{y}
C_{z})

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

In 2004, Cornelsen, a German
textbook company, published a mathematics text entitled **Mathematik
Stochastik: Grun****d- 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.

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"

*art work courtesy of cartalk.cars.com