Solve the Monty Hall Problem using Logic and Mathematics

The Monty Hall Paradox: Image by bzo

Monty Hall, the host of “Let’s Make a Deal”, asks a contestant to choose the one door to a valuable prize, rejecting the other two doors leading to junk. The host then opens one of the rejected doors, revealing a junk prize. The “Monty Hall problem” now is: should the contestant stay with the original choice, or switch to the other unopened door?

Setting the Stage for the Monty Hall Problem

Let’s label the doors ‘A’, ‘B’ and ‘C’, even though the game may have called them ’1′, ’2′ and ’3′. We will see why in the next sub-section.

For a science-oriented experiment, let’s ensure that Monty Hall has no prior knowledge about what is hidden behind which door. That way, he cannot reveal anything to an observant contestant.

While we’re stipulating conditions, let’s also say that there is neither cheating nor ESP: the contestant has no prior knowledge, supernatural insight or extra assistance to help with the selection process.

The final condition is that Monty Hall and his team do not cheat by moving the prizes around after the contestant has made a choice… especially not after the second choice!

To Simplify Discussing the Monty Hall Problem

For the sake of this discussion, the contestant will always choose door ‘A’ first. This convention simplifies the discussion, because ‘A’ really means “the door that the contestant first chose.” Monty Hall will then reveal a door with a junk prize.

Probability Theory, Logic and the Monty Hall Problem

It is obvious that the contestant has a one-third chance of making a correct guess at the start. The Monty Hall solution, however, depends on whether the probability changes after one junk prize has been revealed.

So, the contestant chose door ‘A’. Monty Hall has revealed door ‘C’ with its junk prize. What should the contestant do to maximize the chances of winning the valuable prize? Can math or logic be of more help?

Two Incorrect Approaches to the Monty Hall Problem using Logic

Let’s first approach the Monty Hall problem using logic and reasoning, saving the mathematics for later.

An Immobile Prize Leaves the Odds the Same

The prize did not move, so each door, and the contestant, keep the one-third chance of being correct. According to logic, there is nothing to gain by switching, since each remaining door retains its one-third chance.

The conclusion is wrong. Although the prize did not move, the contestant now has more information: one “losing” door has been eliminated.

The Odds Have Become Even

Now the contestant knows that the prize is behind one of the two remaining doors. Therefore, there is a fifty-fifty chance, or “even odds”, that either door has the valuable prize. Therefore, according to logic, there is no advantage to switching.

This reasoning is also faulty. It treats the situation as if the contestant had not made the first choice at all. In truth, the fact that the contestant has made one choice means that even more information is now available.

A similar fallacy that leads to the correct behaviour accepts the false notion that the odds have become even. A correct conclusion from that fallacy is “switch because now the contestant has a 1/2 rather than 1/3 chance of being correct”.

Correctly Solving the Monty Hall Problem using Logic

The contestant choses door ‘A.’ Monty Hall cannot reveal door ‘A’ without ending the game prematurely. He also cannot reveal the valuable prize, so when Monty Hall opens door ‘C’, it must reveal junk. What choices does the game’s host have?

There are two situations. If the contestant chose ‘A’ correctly the first time, then either ‘B’ or ‘C’ became equally available. The contestant had 1/3 odds that ‘A’ had the valuable prize. As was stated before: sticking to the original choice does not change that chance of success. Thus, the strategy of “always switch” inherits that 1/3 chance… of choosing incorrectly.

In the second situation, the contestant’s original choice of ‘A’ led to junk. If door ‘B’ had the prize, then Monty Hall had no choice but to open door ‘C’. The contestant had a 2/3 chance that door ‘A’ had junk, but the consequence is a 100% certainty that the only other “junk” door would be opened. Therefore, there is a 2/3 chance that switching results in the correct door.

Another correct way of understanding the situation is that the original choice has a 1/3 chance of success. Suppose Monty Hall were to then ask the contestant, “Would you rather switch to select the other two doors, and give me back the prize you don’t want”? A wise contestant would switch to the two other doors, and have the 2/3 chance of success. This is essentially the choice Monty Hall does give the contestant, by opening the non-selected junk prize door.

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?

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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Tags: calculating odds, choosing a door, door number one, door number three, door number two, lets make a deal, logic, marilyn savant, marilyn vos savant, monty hall, monty hall problem, monty hall question, monty hall solution, probability theory