The last word: What are the odds?
A classic quandary inspired by an old TV game show reveals a universal human flaw, says author Leonard Mlodinow. Life is a series of probability decisions, and we
As newspaper columns go, Parade magazine’s “Ask Marilyn” has to be considered a smashing success. Distributed in 350 newspapers and boasting a combined circulation of nearly 36 million, the question-and-answer column originated in 1986 and is still going strong.
Who is this guru Marilyn? Well, Marilyn vos Savant is famous for being listed for years in the Guinness World Records Hall of Fame as the person with the world’s highest recorded IQ (228). She is also famous for being married to Robert Jarvik, inventor of the Jarvik artificial heart. But sometimes famous people, despite their other accomplishments, are remembered for something they wished had never happened. That may be the case for Marilyn, who is most famous for her response to the following question (slightly reworded below), which appeared in her column one Sunday in September 1990:
“Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, ‘Do you want to switch to the other unopened door?’ Is it to the contestant’s advantage to make the switch?”
The question was inspired by the workings of the television game show Let’s Make a Deal, which ran from 1963 to 1976 and in several incarnations from 1980 to 1991. The show’s main draw was its amiable host, Monty Hall, and his beauty queen assistant, Carol Merrill.
It had come to the surprise of the show’s creators that after airing 4,500 episodes in nearly 27 years, it was this question of mathematical probability that would be their principal legacy. This issue has immortalized both Marilyn and Let’s Make a Deal because of the vehemence with which Marilyn vos Savant’s readers responded to the column. After all, it appears to be a pretty silly question. Two doors are available—open one and you win; open the other and you lose—so it seems self-evident that whether you change your choice or not, your chances of winning are 50/50. What could be simpler? The thing is, Marilyn said in her column that it is better to switch.
Despite the public’s much-heralded lethargy when it comes to mathematical issues, Marilyn’s readers reacted as if she’d advocated ceding California back to Mexico. Her denial of the obvious brought her an avalanche of mail, 10,000 letters by her estimate. If you ask the American people whether they agree that plants create the oxygen in the air, light travels faster than sound, or you cannot make radioactive milk by boiling it, you will get double-digit disagreement in each case (13 percent, 24 percent, and 35 percent, respectively). But on this issue, Americans were united: Ninety-two percent agreed Marilyn was wrong.
Almost 1,000 Ph.D.s wrote in, many of them math professors, who seemed especially irate. This came from Georgetown: “How many irate mathematicians are needed to change your mind?” Someone from the U.S. Army Research Institute remarked, “If all those Ph.D.s are wrong the country would be in serious trouble.”
The thing is, Marilyn was correct. When told of this, Paul Erdos, one of the leading mathematicians of the 20th century, said, “That’s impossible.” Then, when presented with a formal mathematical proof of the correct answer, he still didn’t believe it and grew angry. Only after a colleague arranged for a computer simulation in which Erdos watched hundreds of trials that came out 2-to-1 in favor of switching did Erdos concede that he was wrong.
How can something that seems so obvious be wrong? In the words of a Harvard professor who specializes in probability and statistics, “Our brains are just not wired to do probability problems very well.”
That’s a shame. As Roman statesman Cicero wrote, “Probability is the very guide of life.” The outline of our lives, like the flame of a candle, is continuously coaxed in new directions by a variety of random events that, along with our responses to them, determine our fate. Life, as a result, is both hard to predict and hard to interpret. Just as, looking at a Rorschach blot, you might see Madonna and I, a duck-billed platypus, the data we encounter in business, law, medicine, sports, the media, or your child’s third-grade report card can be read in many ways. Yet interpreting the role of chance in an event is not like interpreting a Rorschach blot; there are right ways and wrong ways to do it.
We often employ intuitive processes when we make assessments and choices in uncertain situations. But those intuitive processes come with drawbacks. They no doubt carried an evolutionary advantage when we had to decide whether a saber-toothed tiger was smiling because it was fat and happy or because it was famished and saw us as its next meal. But the modern world has a different balance.
The human response to uncertainty is so complex that different structures within the brain can come to separate conclusions. For example, if your face swells to five times its normal size three out of four times you eat shrimp, the “logical” left hemisphere of your brain will attempt to find a pattern. The “intuitive” right hemisphere, on the other hand, will simply say, “Avoid shrimp.” Sometimes the separate hemispheres apparently fight it out to determine which conclusion will dominate.
At least that’s what researchers found in less painful experimental setups. In a game called probability guessing, subjects are shown a series of cards or lights, which can have two colors, say green and red. Things are arranged so that the colors will appear with different probabilities but otherwise without a pattern. For example, red might appear twice as often as green. The task of the subject, after watching for a while, is to predict whether each new member of the sequence will be red or green.
The game has two basic strategies. One is to always guess the color that you notice occurs more frequently. That is the route favored by rats and other (nonhuman) animals. The other strategy is to “match” your proportion of green and red guesses to the proportion of green and red you observed in the past.
Humans usually adapt the second strategy, trying to deduce a pattern. In the process we allow ourselves to be outperformed by a rat. But there are people with certain types of post-surgical brain impairment—called a split brain—that precludes the right and left hemispheres from communicating with each other. When the experiment is performed on these people in such a way that the separate hemispheres of each subject are tested separately, the right hemisphere always chooses to guess the more frequent color and the left hemisphere always tries to guess the pattern.
To humans, random events often look like non-random events. We must be careful, when interpreting human affairs, to never confuse the two.
The Monty Hall problem can be solved without any specialized mathematical knowledge, but you do need a basic understanding of how probability truly works. The key is the law of the “sample space”—a framework for analyzing chance situations that was first put on paper in the 16th century by an Italian mathematician and gambler named Gerolamo Cardano.
In modern language, Cardano’s rule reads like this: “Suppose a random process has many equally likely outcomes, some favorable, some unfavorable. Then the probability of obtaining a favorable outcome is equal to the proportion of outcomes that are favorable.”
The potency of Cardano’s rule goes hand in hand with certain subtleties. One lies in the meaning of the term “outcomes.” As late as the 18th century, the famous French mathematician Jean Le Rond d’Alembert misused the concept when he analyzed the toss of two coins. The number of heads that turns up in those two tosses can be 0, 1, or 2. Since there are three outcomes, Alembert reasoned, the chances of each must be 1 in 3. But Alembert was mistaken.
The key is to realize that the possible outcomes of each of the two coin tosses must be considered. In other words, we have to think about the tosses in terms of possible sequences (heads, heads), (heads, tails), (tails, heads), and (tails, tails). These are the four possibilities that make up the “sample space.” There is thus a 75 percent chance that the two tosses will yield at least one heads but only a 25 percent chance that they will yield two.
In the Monty Hall problem, you are facing three doors, and the sample space is this list of three possible favorable outcomes:
The car is behind door 1.
The car is behind door 2.
The car is behind door 3.
When you first chose door 1, each of these outcomes had a probability of 1 in 3. But according to the problem, the next thing that happens is that the host, who knows what’s behind all the doors, opens one you did not choose, revealing one of the goats. In opening this door, the host has used what he knows to avoid revealing the car, so this is not a completely random process.
There are two cases to consider. One is the case in which your initial choice was correct. Let’s call that the Lucky Guess scenario. The other is the case in which your initial choice was wrong. The chances your initial choice was correct are 1 out of 3, and the chances you guessed wrong are 2 out of 3, so the Wrong Guess scenario is twice as likely as the Lucky Guess scenario. In the Lucky Guess scenario, the host randomly opens one of the doors you did not choose and you will win if you stick with your door. In the Wrong Guess scenario, the host does not want to reveal the car, so he chooses to open the one door available to him that does not have a car behind it, and you will win if you switch doors.
Your decision thus comes down to a guess: In which scenario do you find yourself? If you feel that ESP or fate has guided your initial choice, maybe you shouldn’t switch. But unless you can bend silver spoons into pretzels with your brain waves, the odds are 2-to-1 that you are in the Wrong Guess Scenario.
And so, it is better to switch.
From The Drunkard’s Walk by Leonard Mlodinow. ©2008 by Leonard Mlodinow. Used by permission of Pantheon Books.