Conjunction Fallacy: Why Detail Makes a Story Feel More Likely

The rule being broken is the conjunction rule of probability: the chance that A and B are both true can never exceed the chance that A is true, because every world in which Linda is a feminist bank teller is a world in which she is a bank teller. Most people accept the rule the moment it is stated. The Linda problem shows that they do not apply it when they judge, because they are not computing probabilities at all. They are judging resemblance.

Tversky and Kahneman's explanation was the representativeness heuristic. Asked how probable a statement is, people ask themselves how well it fits the picture they have formed. Linda's description was written to fit an active feminist and to clash with a bank teller. Adding the feminist clause to the bank teller statement makes the whole thing more like Linda, so it feels more probable, even though logically it has become a narrower claim. In the same paper, 88 students asked outright how well each statement resembled Linda ranked the conjunction above the bank teller statement 85 percent of the time, and the probability rankings followed the resemblance rankings.

Two things make the error durable. Training does not reliably remove it: Tversky and Kahneman found violations in graduate students who had taken statistics and in doctoral students in decision science as well as in undergraduates with no background. Transparency does not remove it either: when only the two critical statements were shown side by side, 85 percent still chose the conjunction. The better story pulls even on people who know the rule.

The size of the effect depends on the question, and critics have argued about what that means. Hertwig and Gigerenzer showed in 1999 that participants often read the word probable in its everyday senses, as plausible or believable, and that asking instead how many of 100 women like Linda are bank tellers cut the error sharply; Fiedler had reported a similar drop with frequency wording in 1988. Tversky and Kahneman had found the same with a health survey problem in their own paper, described below. The fallacy is robust in the probability wording; how much of it is a reasoning error and how much a reading of the question is the live debate.

Amos Tversky and Daniel Kahneman published "Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment" in Psychological Review in 1983. Linda is the problem everybody remembers.

Participants read a short description: Linda is 31 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 anti-nuclear demonstrations. They then ranked eight statements about her by probability. Five were fillers (elementary school teacher, bookstore worker who takes yoga classes, psychiatric social worker, member of the League of Women Voters, insurance salesperson). The three that mattered were that Linda is active in the feminist movement, that she is a bank teller, and that she is a bank teller and is active in the feminist movement.

Three groups took part: statistically naive undergraduates at Stanford and the University of British Columbia, graduate and medical students with at least one statistics course, and doctoral students in decision science at Stanford's business school. Among the naive group, 89 percent ranked the feminist bank teller above the bank teller; the informed group did no better, at 90 percent. The authors then tried what they called increasingly desperate manipulations to get people to obey the rule. In the most transparent, 142 undergraduates saw only the two statements and checked which was more probable. 85 percent chose the conjunction.

The paper also reports what helped. In the health survey problem, asking for percentages produced errors from 65 percent of 147 respondents; asking how many of 100 men had had a heart attack, and how many were over 55 and had had one, reduced the error to 25 percent of 117; asking first how many of the 100 were over 55 brought it to 11 percent. The authors read this as evidence that people reason extensionally when the problem makes the classes visible, and judge by resemblance when it does not.

The fallacy appears wherever a specific scenario competes with the broader class it belongs to. The rows below are illustrations, not findings from the studies cited here.

The counters follow the manipulations that worked in the 1983 paper and in the later frequency studies: make the classes visible and count them.

• Translate probability into frequency. Instead of asking how likely it is that Linda is a feminist bank teller, ask how many of 100 women fitting her description are bank tellers, and how many of those are also feminists. The second number cannot exceed the first.
• Rate the broad category before the specific case. If you have already said the chance of a recession is 20 percent, you will notice when you are about to give a banking-crisis recession 30.
• Treat added detail as a cost, not a credential. Every clause an account adds is a further condition that must be true, and the probability can only go down.
• Separate plausibility from probability. A story can fit the facts beautifully and still be less likely than a duller story that covers more cases.
• Draw the nest. A circle for bank tellers with a smaller circle inside it for feminist bank tellers keeps the rule in view.

The conjunction fallacy belongs to a family of errors that Tversky and Kahneman traced to judging by resemblance. Its neighbors break different rules.

• It is not the base rate fallacy, which ignores how common a category is in the population when judging a case. The conjunction fallacy ignores the relationship between a category and its subset.
• It is not the gambler's fallacy, which concerns sequences of independent events and the belief that a run must correct itself. The conjunction fallacy concerns a single case and two descriptions of it.
• It is not insensitivity to sample size, the failure to see that small samples vary more than large ones. That error is about variability; this one is about set inclusion.
• It is not the neglect of probability, in which the likelihood of an outcome is set aside altogether in favor of its vividness. In the Linda problem people are trying to judge probability; they are judging the wrong thing.