Although probability theory may seem simple and very intuitive, it can be very subtle and deceptive.

**THE MONTY HALL PROBLEM**

Although probability theory may
seem simple and very intuitive, it can be very subtle and deceptive. Many
results found in the field of probability are counterintuitive; some examples are the St. Petersburg Paradox,
Benford’s Law of Lead Digits, the Birthday Problem, Simpson’s Paradox, and the
Monty Hall Problem.

References for further reading on the foregoing
problems include Feller (1971), which provides a good treatment of Benford’s
Law, the Waiting Time Paradox, and the Birthday Problem. We also recommend the
delightful account (Bruce, 2000), written in the style of Arthur Conan Doyle,
wherein Sherlock Holmes teaches Wat-son about many probability misconceptions.
Simpson’s Paradox, which is impor-tant in the analysis of categorical data in
medical studies, will be addressed in Chap-ter 11.

The Monty Hall Problem achieved fame and notoriety
many years ago. Marilyn Vos Savant, in her *Parade*
magazine column, presented a solution to the problem in response to a reader’s
question. There was a big uproar; many readers responded in writing (some in a very
insulting manner), challenging her answer. Many of those who offered the
strongest challenges were mathematicians and statisticians. Never-theless, Vos
Savant’s solution, which was essentially correct, can be demonstrated easily
through computer simulation.

In the introduction to her book (1997), Vos Savant
summarizes this problem, which she refers to as the Monty Hall Dilemma, as well
as her original answer. She repeats this answer on page 5 of the text, where
she discusses the problem in more detail and provides many of the readers’
written arguments against her solution.

On pages 5–17, she presents the succession of
responses and counterresponses. Also included in Vos Savant (Appendix, pages
169–196) is Donald Granberg’s well-formulated and objective treatment of the
mathematical problem. Granberg provides insight into the psychological
mechanisms that cause people to cling to in-correct answers and not consider
opposing arguments. Vos Savant (1997) is also a good source for other
statistical fallacies and misunderstandings of probability.

The Monty Hall Problem may be stated as follows: At
the end of each “Let’s Make a Deal” television program, Monty Hall would let
one of the contestants from that episode have a shot at the big prize. There
were three showcase doors to choose from. One of the doors concealed the prize,
and the other two concealed “clunkers” (worthless prizes sometimes referred to
as “goats”).

In fact, a real goat actually might be standing on
the stage behind one of the doors! Monty would ask a contestant to choose a
door; then he would expose one of the other doors that was hiding a clunker.
Then the contestant would be offered a bribe ($500, $1000, or more) to give up
the door. Generally, the contestants chose to keep the door, especially if
Monty offered a lot of cash for the bribe; the grand prize was always worth a
lot more than the bribe. The more Monty offered, the more the contestants
suspected that they had the right door. Since Monty knew which door held the
grand prize, contestants suspected that he was tempting them to give up the
grand prize.

The famous problem that Vos Savant addressed in her
column was a slight vari-ation, which Monty may or may not have actually used.
Again, after one of the three doors is removed, the contestant selects one of
the two remaining doors. In-stead of offering money, the host (for example,
Monty Hall) allows the contestant to keep the selected door or switch to the
remaining door. Marilyn said that the contestant should switch because his chance
of winning if he switches is 2/3, while the door he originally chose has only a
1/3 chance of being the right door.

Those who disagreed said that it would make no
difference whether or not the contestant switches, as the removal of one of the
empty doors leaves two doors, each with an equal 1/2 chance of being the right
door. To some, this seemed to be a simple exercise in conditional
probabilities. But they were mistake*n*!

One correct argument would be that initially one
has a 1/3 chance of selecting the correct door. Once a door is selected, Monty
will reveal a door that hides a clunker. He can do this only because he knows
which door has the prize. If the first door selected is the winner, Monty is
free to select either of the two remaining doors. However, if the contestant
does not have the correct door, Monty must show the contestant the one
remaining door that conceals a clunker.

But the correct door will be found two-thirds of
the time using a switching strat-egy. So in two-thirds of the cases, switching
is going to lead one to the winning door; only in one-third of the cases will
switching backfire. Consequently, a strate-gy of always switching will win
about 67% of the time, and a strategy of remaining with the selected door will
win only 33% of the time.

Some of the mathematicians erred because they
ignored the fact that the contes-tant picked a door first, thus affecting
Monty’s strategy. Had Monty picked one of the two “clunker” doors first at
random, the problem would be different. The con-testant then would know that
each of the two remaining doors has an equal (50%) chance of being the right
door. Then, regardless of which door the contestant chose, the opportunity to
switch would not affect the chance of winning: 50% if he stays, and 50% if he switches.
The subtlety here is that the difference in the order of the decisions
completely changes the game and the probability of the final outcome.

If you still do not believe that switching doubles
your chances of winning, con-struct the game on a computer. Use a uniform
random number generator to pick the winning door and let the computer follow
Monty’s rule for showing a clunker door. That is the best way to see that after
playing the game many times (e.g., at least 100 times employing the switching strategy
and 100 times employing the staying strate-gy), you will win nearly 67% of the
time when you switch and only about 33% of the time when you keep the same
door.

If you are not adept at computer programming, you
can go to the Susan Holmes Web site at the Stanford University Statistics
Department (www.stat. stanford.edu/~susan). She has a computerized version of
the game that you can play; she will keep a tally of the number of wins out of
the number of times you switch and also a tally of the number of wins out of
the number of times you re-main with your first choice.

The game works as follows: Susan shows you a
cartoon with three doors. First, you click on the door you want. Next, her
computer program uncovers a door showing a cartoon picture of a donkey. Again,
you click on your door if you want to keep it or click on the remaining door if
you want to switch. In response, the program shows you what is behind your
door: either you win or find another donkey.

Then you are asked if you want to play again. You
can play the game as many times as you like using whatever strategy you like.
Finally, when you decide to stop, the program shows you how many times you won
when you switched and the total number of times you switched. The program also
tallies the number of times you won when you used the staying strategy, along
with the total number of times you chose this strategy.

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