Solve the Monty Hall Problem using Logic and Mathematics

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The Math Proof to Enumerate the Monty Hall Problem

This problem is small enough to be resolved by listing the possibilities; this is one valid way to prove the mathematics of probability theory. Please refer to the chart above.

There are three initial setup conditions, numbered #1, #2 and #3. These correspond to doors ‘A’, ‘B’ and ‘C’ being either “Junk” or “Valuable”. The contestant chooses ‘A’ in each setup. Monty Hall then reveals a door.

The contestant has already decided on a strategy, and so is not influenced by exactly which door Monty Hall opens.

Marilyn vos Savant was Ridiculed for Solving the Monty Hall Problem

In 1990, Marilyn vos Savant was asked to solve the Monty Hall problem. She gave the correct strategy, and was promptly ridiculed by many fans of the show. Among them were mathematicians who said the probability jumped from 1/3 to 1/2, but not to 2/3.

Who is Marilyn vos Savant?

Marilyn vos Savant hails from St. Louis, Missouri. Known for her extremely high IQ scores in childhood and as an adult, she is a columnist and author. She is also an executive with Jarvik Heart, Inc., a company that develops and manufactures artificial hearts.

Monty Hall’s Response to Marilyn vos Savant

Apparently the rules permitted Monty Hall three twists beyond the assumptions in the first paragraph.

  • First, the host was not obliged to open a “junk” door and offer a switch. He could simply allow the contestant to lose immediately. (This might have allowed the show to close on time).
  • Secondly, Monty Hall could offer a cash incentive to persuade the contestant to switch, whether or not a “junk” door had been revealed. The psychology of resisting such a temptation is well beyond this mathematics article.
  • Finally, the host’s knowledge of exactly what was behind each door gave him enormous power to psychologically manipulate the contestant.

Monty Hall noted that Marilyn vos Savant had solved his problem from a science and mathematics viewpoint, but had not noticed all the nuances of the television program’s rules.

Who is Monty Hall?

Picture of Monty Hall Courtesy of ABC Television

Monty Hall was named “Mauric Halpin” on the occasion of his birth in Winnipeg on Aug. 25th, 1921. A graduate of the University of Manitoba, Mr. Halpin was employed by the Canada Wheat Board before launching his career in television.

Hall is most famous for hosting “Let’s Make a Deal,” where he also was producer and executive producer. Hall had production duties at various levels for “Split Second,” “It’s Anybody’s Guess,” “Masquerade Party,” “Split Second,” “Talking Pictures,” “Your First Impression,” and three episodes of “The McLean Stevenson Show”.

He also acted in, or hosted, several other television programs.

Monty Hall is a member of the Order of Canada, in recognition for his work with Variety Clubs International and the Muscular Dystrophy Association.

References:
Monty Hall. Internet Movie Database. Accessed Aug. 21, 2011.
Nakhoda, Aadil and  Friedman, Dr. Dan. “Monty Hall Problem”. University of California, Santa Cruz. Accessed Aug. 21, 2011.
vos Savant, M. “About Marilyn“. Accessed Aug. 15, 2011.
Weisstein, E. “Monty Hall Problem” MathWorld, a Wolfram Web Resource. Accessed Aug. 21, 2011.

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