Amos and I then collected answers to the same question from 114 graduate students in psychology at three major universities, all of whom had taken several courses in statistics. They did not disappoint us. Their rankings of the nine fields by probability did not differ from ratings by similarity to the stereotype. Substitution was perfect in this case: there was no indication that the participants did anything else but judge representativeness. The question about probability (likelihood) was difficult, but the question about similarity was easier, and it was answered instead. This is a serious mistake, because judgments of similarity and probak tbility are not constrained by the same logical rules. It is entirely acceptable for judgments of similarity to be unaffected by base rates and also by the possibility that the description was inaccurate, but anyone who ignores base rates and the quality of evidence in probability assessments will certainly make mistakes.
The concept “the probability that Tom W studies computer science” is not a simple one. Logicians and statisticians disagree about its meaning, and some would say it has no meaning at all. For many experts it is a measure of subjective degree of belief. There are some events you are sure of, for example, that the sun rose this morning, and others you consider impossible, such as the Pacific Ocean freezing all at once. Then there are many events, such as your next-door neighbor being a computer scientist, to which you assign an intermediate degree of belief—which is your probability of that event.
Logicians and statisticians have developed competing definitions of probability, all very precise. For laypeople, however, probability (a synonym of likelihood in everyday language) is a vague notion, related to uncertainty, propensity, plausibility, and surprise. The vagueness is not particular to this concept, nor is it especially troublesome. We know, more or less, what we mean when we use a word such as democracy or beauty and the people we are talking to understand, more or less, what we intended to say. In all the years I spent asking questions about the probability of events, no one ever raised a hand to ask me, “Sir, what do you mean by probability?” as they would have done if I had asked them to assess a strange concept such as globability. Everyone acted as if they knew how to answer my questions, although we all understood that it would be unfair to ask them for an explanation of what the word means.
People who are asked to assess probability are not stumped, because they do not try to judge probability as statisticians and philosophers use the word. A question about probability or likelihood activates a mental shotgun, evoking answers to easier questions. One of the easy answers is an automatic assessment of representativeness—routine in understanding language. The (false) statement that “Elvis Presley’s parents wanted him to be a dentist” is mildly funny because the discrepancy between the images of Presley and a dentist is detected automatically. System 1 generates an impression of similarity without intending to do so. The representativeness heuristic is involved when someone says “She will win the election; you can see she is a winner” or “He won’t go far as an academic; too many tattoos.” We rely on representativeness when we judge the potential leadership of a candidate for office by the shape of his chin or the forcefulness of his speeches.
Although it is common, prediction by representativeness is not statistically optimal. Michael Lewis’s bestselling Moneyball is a story about the inefficiency of this mode of prediction. Professional baseball scouts traditionally forecast the success of possible players in part by their build and look. The hero of Lewis’s book is Billy Beane, the manager of the Oakland A’s, who made the unpopular decision to overrule his scouts and to select players by the statistics of past performance. The players the A’s picked were inexpensive, because other teams had rejected them for not looking the part. The team soon achieved excellent results at low cost.
The Sins of Representativeness
Judging probability byals representativeness has important virtues: the intuitive impressions that it produces are often—indeed, usually—more accurate than chance guesses would be.
On most occasions, people who act friendly are in fact friendly.
A professional athlete who is very tall and thin is much more likely to play basketball than football.
People with a PhD are more likely to subscribe to The New York Times than people who ended their education after high school.
Young men are more likely than elderly women to drive aggressively.
In all these cases and in many others, there is some truth to the stereotypes that govern judgments of representativeness, and predictions that follow this heuristic may be accurate. In other situations, the stereotypes are false and the representativeness heuristic will mislead, especially if it causes people to neglect base-rate information that points in another direction. Even when the heuristic has some validity, exclusive reliance on it is associated with grave sins against statistical logic.
One sin of representativeness is an excessive willingness to predict the occurrence of unlikely (low base-rate) events. Here is an example: you see a person reading The New York Times on the New York subway. Which of the following is a better bet about the reading stranger?
She has a PhD.
She does not have a college degree.
Representativeness would tell you to bet on the PhD, but this is not necessarily wise. You should seriously consider the second alternative, because many more nongraduates than PhDs ride in New York subways. And if you must guess whether a woman who is described as “a shy poetry lover” studies Chinese literature or business administration, you should opt for the latter option. Even if every female student of Chinese literature is shy and loves poetry, it is almost certain that there are more bashful poetry lovers in the much larger population of business students.
People without training in statistics are quite capable of using base rates in predictions under some conditions. In the first version of the Tom W problem, which provides no details about him, it is obvious to everyone that the probability of Tom W’s being in a particular field is simply the base rate frequency of enrollment in that field. However, concern for base rates evidently disappears as soon as Tom W’s personality is described.
Amos and I originally believed, on the basis of our early evidence, that base-rate information will always be neglected when information about the specific instance is available, but that conclusion was too strong. Psychologists have conducted many experiments in which base-rate information is explicitly provided as part of the problem, and many of the participants are influenced by those base rates, although the information about the individual case is almost always weighted more than mere statistics. Norbert Schwarz and his colleagues showed that instructing people to “think like a statistician” enhanced the use of base-rate information, while the instruction to “think like a clinician” had the opposite effect.
An experiment that was conducted a few years ago with Harvard undergradut oates yielded a finding that surprised me: enhanced activation of System 2 caused a significant improvement of predictive accuracy in the Tom W problem. The experiment combined the old problem with a modern variation of cognitive fluency. Half the students were told to puff out their cheeks during the task, while the others were told to frown. Frowning, as we have seen, generally increases the vigilance of System 2 and reduces both overconfidence and the reliance on intuition. The students who puffed out their cheeks (an emotionally neutral expression) replicated the original results: they relied exclusively on representativeness and ignored the base rates. As the authors had predicted, however, the frowners did show some sensitivity to the base rates. This is an instructive finding.
When an incorrect intuitive judgment is made, System 1 and System 2 should both be indicted. System 1 suggested the incorrect intuition, and System 2 endorsed it and expressed it in a judgment. However, there are two possible reasons for the failure of System 2—ignorance or laziness. Some people ignore base rates because they believe them to be irrelevant in the presence of individual information. Others make the same mist
ake because they are not focused on the task. If frowning makes a difference, laziness seems to be the proper explanation of base-rate neglect, at least among Harvard undergrads. Their System 2 “knows” that base rates are relevant even when they are not explicitly mentioned, but applies that knowledge only when it invests special effort in the task.
The second sin of representativeness is insensitivity to the quality of evidence. Recall the rule of System 1: WYSIATI. In the Tom W example, what activates your associative machinery is a description of Tom, which may or may not be an accurate portrayal. The statement that Tom W “has little feel and little sympathy for people” was probably enough to convince you (and most other readers) that he is very unlikely to be a student of social science or social work. But you were explicitly told that the description should not be trusted!
You surely understand in principle that worthless information should not be treated differently from a complete lack of information, but WY SIATI makes it very difficult to apply that principle. Unless you decide immediately to reject evidence (for example, by determining that you received it from a liar), your System 1 will automatically process the information available as if it were true. There is one thing you can do when you have doubts about the quality of the evidence: let your judgments of probability stay close to the base rate. Don’t expect this exercise of discipline to be easy—it requires a significant effort of self-monitoring and self-control.
The correct answer to the Tom W puzzle is that you should stay very close to your prior beliefs, slightly reducing the initially high probabilities of well-populated fields (humanities and education; social science and social work) and slightly raising the low probabilities of rare specialties (library science, computer science). You are not exactly where you would be if you had known nothing at all about Tom W, but the little evidence you have is not trustworthy, so the base rates should dominate your estimates.
How to Discipline Intuition
Your probability that it will rain tomorrow is your subjective degree of belief, but you should not let yourself believe whatever comes to your mind. To be useful, your beliefs should be constrained by the logic of probability. So if you believe that there is a 40% chance plethat it will rain sometime tomorrow, you must also believe that there is a 60% chance it will not rain tomorrow, and you must not believe that there is a 50% chance that it will rain tomorrow morning. And if you believe that there is a 30% chance that candidate X will be elected president, and an 80% chance that he will be reelected if he wins the first time, then you must believe that the chances that he will be elected twice in a row are 24%.
The relevant “rules” for cases such as the Tom W problem are provided by Bayesian statistics. This influential modern approach to statistics is named after an English minister of the eighteenth century, the Reverend Thomas Bayes, who is credited with the first major contribution to a large problem: the logic of how people should change their mind in the light of evidence. Bayes’s rule specifies how prior beliefs (in the examples of this chapter, base rates) should be combined with the diagnosticity of the evidence, the degree to which it favors the hypothesis over the alternative. For example, if you believe that 3% of graduate students are enrolled in computer science (the base rate), and you also believe that the description of Tom W is 4 times more likely for a graduate student in that field than in other fields, then Bayes’s rule says you must believe that the probability that Tom W is a computer scientist is now 11%. If the base rate had been 80%, the new degree of belief would be 94.1%. And so on.
The mathematical details are not relevant in this book. There are two ideas to keep in mind about Bayesian reasoning and how we tend to mess it up. The first is that base rates matter, even in the presence of evidence about the case at hand. This is often not intuitively obvious. The second is that intuitive impressions of the diagnosticity of evidence are often exaggerated. The combination of WY SIATI and associative coherence tends to make us believe in the stories we spin for ourselves. The essential keys to disciplined Bayesian reasoning can be simply summarized:
Anchor your judgment of the probability of an outcome on a plausible base rate.
Question the diagnosticity of your evidence.
Both ideas are straightforward. It came as a shock to me when I realized that I was never taught how to implement them, and that even now I find it unnatural to do so.
Speaking of Representativeness
“The lawn is well trimmed, the receptionist looks competent, and the furniture is attractive, but this doesn’t mean it is a well-managed company. I hope the board does not go by representativeness.”
“This start-up looks as if it could not fail, but the base rate of success in the industry is extremely low. How do we know this case is different?”
“They keep making the same mistake: predicting rare events from weak evidence. When the evidence is weak, one should stick with the base rates.”
“I know this report is absolutely damning, and it may be based on solid evidence, but how sure are we? We must allow for that uncertainty in our thinking.”
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Linda: Less Is More
The best-known and most controversial of our experiments involved a fictitious lady called Linda. Amos and I made up the Linda problem to provide conclusive evidence of the role of heuristics in judgment and of their incompatibility with logic. This is how we described Linda:
Linda is thirty-one years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.
The audiences who heard this description in the 1980s always laughed because they immediately knew that Linda had attended the University of California at Berkeley, which was famous at the time for its radical, politically engaged students. In one of our experiments we presented participants with a list of eight possible scenarios for Linda. As in the Tom W problem, some ranked the scenarios by representativeness, others by probability. The Linda problem is similar, but with a twist.
Linda is a teacher in elementary school.
Linda works in a bookstore and takes yoga classes.
Linda is active in the feminist movement.
Linda is a psychiatric social worker.
Linda is a member of the League of Women Voters.
Linda is a bank teller.
Linda is an insurance salesperson.
Linda is a bank teller and is active in the feminist movement.
The problem shows its age in several ways. The League of Women Voters is no longer as prominent as it was, and the idea of a feminist “movement” sounds quaint, a testimonial to the change in the status of women over the last thirty years. Even in the Facebook era, however, it is still easy to guess the almost perfect consensus of judgments: Linda is a very good fit for an active feminist, a fairly good fit for someone who works in a bookstore and takes yoga classes—and a very poor fit for a bank teller or an insurance salesperson.
Now focus on the critical items in the list: Does Linda look more like a bank teller, or more like a bank teller who is active in the feminist movement? Everyone agrees that Linda fits the idea of a “feminist bank teller” better than she fits the stereotype of bank tellers. The stereotypical bank teller is not a feminist activist, and adding that detail to the description makes for a more coherent story.
The twist comes in the judgments of likelihood, because there is a logical relation between the two scenarios. Think in terms of Venn diagrams. The set of feminist bank tellers is wholly included in the set of bank tellers, as every feminist bank teller is0%"ustwora ban0%" w a bank teller. Therefore the probability that Linda is a feminist bank teller must be lower than the probability of her being a bank teller. When you specify a possible event in greater detail you can only lower its probability. The problem therefore sets up a conflict between the intuition of representativeness and the logic of probability.
Our initia
l experiment was between-subjects. Each participant saw a set of seven outcomes that included only one of the critical items (“bank teller” or “feminist bank teller”). Some ranked the outcomes by resemblance, others by likelihood. As in the case of Tom W, the average rankings by resemblance and by likelihood were identical; “feminist bank teller” ranked higher than “bank teller” in both.
Then we took the experiment further, using a within-subject design. We made up the questionnaire as you saw it, with “bank teller” in the sixth position in the list and “feminist bank teller” as the last item. We were convinced that subjects would notice the relation between the two outcomes, and that their rankings would be consistent with logic. Indeed, we were so certain of this that we did not think it worthwhile to conduct a special experiment. My assistant was running another experiment in the lab, and she asked the subjects to complete the new Linda questionnaire while signing out, just before they got paid.
Thinking, Fast and Slow Page 18
