The Monty Hall Problem

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 mistaken!

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.