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Conjunction fallacy

type: effect · origin: study · evidence: supported · themes: judgements and probabilities

Claim assessed: People judge a detailed story as more likely than one of its parts.

Published · how we work

In short

The conjunction fallacy is judging two things together to be more probable than one of them alone. The rule is simple: “A and B” can never be more probable than “A”, because every case of “A and B” is also a case of “A”. Yet when the added detail fits a story, it seems to make the story more believable. Tversky and Kahneman (1983) showed the error in students, in statistics experts and in physicians. Later work showed that it depends a great deal on the wording and on the story used.

What it says

The “Linda” problem. Tversky and Kahneman gave participants this description: Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with discrimination and social justice, and took part in anti-nuclear demonstrations. They then asked them to rank several statements by probability, among them “Linda is a bank teller” and “Linda is a bank teller and is active in the feminist movement”. Between 85% and 90% ranked the second statement above the first. The proportion was the same among students with no training in statistics and among decision science doctoral students at Stanford, who had taken advanced courses in probability.

Why. The authors propose that people don’t judge probability but how well the description matches a type of person. Another group ranked the same statements by how much Linda resembles the “typical member” of each category. The correlation between the ranking by resemblance and the ranking by probability was 0.98. The mechanism is called the representativeness heuristic.

The attempts to make it go away. The authors describe what followed as a series of “increasingly desperate manipulations”:

Not only Linda. The same pattern appeared among 103 internists, who judged possible symptoms in a patient. On average, 91% judged a combination of a typical and an atypical symptom more probable than the atypical symptom alone. At a forecasting conference in 1982, one group of professional analysts estimated the probability of a break in diplomatic relations between the USA and the USSR. Another group estimated the probability that the USSR would invade Poland and relations would break down. The second group gave higher estimates, although its scenario was strictly narrower. A plausible cause makes a story more believable, but less probable.

Thinking in numbers of cases. In a problem about a health survey, 65% of students estimated that the percentage of men who “are over 55 and have had a heart attack” was higher than the percentage who “have had a heart attack”. When they were first asked for the percentage over 55, the error dropped to 31%. When the question was “how many of 100 participants”, it dropped to 25%, and with both changes, to 11%.

The dispute about words. Hertwig and Gigerenzer (1999) argued that many participants don’t understand “probability” in its mathematical sense but in one of the word’s other senses. In their view, the word “frequency” imposes the mathematical sense, which is why errors drop. Hertwig also proposed that “and” can be read as “or”, as in “we invited friends and colleagues”. Kahneman and Hertwig tested these ideas together, with Barbara Mellers as arbiter (Mellers et al., 2001). Each group of participants saw a single version and estimated how many of 100 people like Linda would be in that category. For Linda, the error appeared with “and” in all three experiments. With the stricter wording “and are”, it disappeared in one experiment and appeared in the other two. With a second story, about a man called James, it was absent in the first experiment. The authors remained in disagreement about the interpretation.

The replication. Chandrashekar et al. (2021) repeated the study with 1,032 participants, with the hypotheses registered in advance. For Linda, the error appeared clearly, and “and” and “and are” gave the same result, so the wording made no difference. For James, the error did not appear at all.

In a 2021 replication with 1,032 participants, each group estimated how many of 100 people like Linda or James are in a category. For Linda: feminists 58.4, bank tellers 9.9, bank tellers and feminists 18.8, bank tellers and are feminists 19.6, so the combination seemed more frequent than part of it. For James: artists 36.2, Republicans 18.4, Republicans and artists 15.2, Republicans and are artists 15.6, so here the error did not appear. Linda and James a replication with 1,032 participants (Chandrashekar et al., 2021) Of 100 people like her or him, how many are... Linda feminists 58.4 bank tellers 9.9 bank tellers and feminists 18.8 bank tellers and are feminists 19.6 James artists 36.2 Republicans 18.4 Republicans and artists 15.2 Republicans and are artists 15.6 For Linda, the combination seemed almost twice as frequent as part of it. For James, the error did not appear.
Chandrashekar et al. (2021), Table 1: mean estimates, with 252 to 260 participants per group. Each group saw a single category for each story.

Stakes and discussion. Charness, Karni and Levin (2010) report, in their abstract, that when the correct answer carried a small reward, the proportion of people who got Linda wrong was significantly lower than the one Tversky and Kahneman reported. It fell sharply when participants could talk it over in groups, especially in groups of three. Yechiam and Zeif (2023) combined 11 studies with 3,276 participants. Rewards raised the chance of a correct answer, but the effect was small, d = 0.19.

Example

A founder presents a plan: “We launch the app in March, get funding in May, hire two developers by June and reach 10,000 users by December.” Each step sounds reasonable, and the whole story sounds convincing. But the plan succeeds only if every step does, so it is less probable than any single step. The example is an illustration, not a study cited here.

How to apply it

The steps below are a practical way of applying it that we propose, based on the studies cited.

  1. When a story gets more detailed, ask whether it became more probable or just more convincing. Every detail is one more condition.
  2. Think in numbers of cases. “Out of 100 situations like this, in how many does it happen?” In Tversky and Kahneman’s study, the change cut the error from 65% to 25%.
  3. Estimate the broad category first. Ask how probable “A” alone is before judging “A and B”. In the same study, this step cut the error from 65% to 31%, and together with numbers of cases, to 11%.
  4. Watch out for plausible causes. An added motive or mechanism makes a scenario easier to believe, not more probable. The same goes for a plan with many steps: the chance of the whole is lower than that of its weakest step.
  5. Talk it over with someone. In Charness et al.’s study, small groups got it wrong much less often than people answering alone.

Limits and nuances

Sources

See also: Availability heuristic, Hot hand fallacy, Planning fallacy