The Conjunction FallacyAlias: The Conjunction Effect
Type: Probabilistic Fallacy
Which of the following events is most likely to occur within the next year:
The probability of a conjunction is never greater than the probability of its conjuncts. In other words, the probability of two things being true can never be greater than the probability of one of them being true, since in order for both to be true, each must be true. However, when people are asked to compare the probabilities of a conjunction and one of its conjuncts, they sometimes judge that the conjunction is more likely than one of its conjuncts. This seems to happen when the conjunction suggests a scenario that is more easily imagined than the conjunct alone.
Interestingly, psychologists Kahneman and Tversky discovered in their experiments that statistical sophistication made little difference in the rates at which people committed the conjunction fallacy. This suggests that it is not enough to teach probability theory alone, but that people need to learn directly about the conjunction fallacy in order to counteract the strong psychological effect of imaginability.
One of Kahneman and Tversky's tests is known as "the Linda Problem":
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.
Is it more likely that Linda is a bank teller, or a bank teller and feminist? Of course, it is more likely that she is the conjunct than the conjunction. However, the description of Linda given in the problem fits the stereotype of a feminist, whereas it doesn't fit the stereotypical bank teller. Because it is easy to imagine Linda as a feminist, people may misjudge that she is more likely to be both a bank teller and a feminist than a bank teller.
Source: Amos Tversky & Daniel Kahneman, "Judgments of and by Representativeness", in Judgment Under Uncertainty: Heuristics and Biases, Kahneman, Paul Slovic, and Tversky, editors (1985), pp. 84-98.
1 is more probable than 2. No matter how unlikely it is that all American troops will be withdrawn from Iraq within a year, it is less likely that this will happen and that the U.S. will bomb Iranian nuclear facilities.
Here is a proof of the theorem of probability theory that a conjunction is never more probable than its conjuncts. For the axioms cited, see the entry for Probabilistic Fallacy.
Theorem: P(s & t) ≤ P(s)
Proof: By Axiom 4 and the fact that P(s & t) = P(t & s), it follows that P(s & t) = P(t | s)P(s). Now, 0 ≤ P(t | s) ≤ 1, by Axiom 1 and the fact that P(s) ≤ 1, for all s. The theorem follows from a general fact about inequalities: if a = bc and 0 ≤ b ≤ 1, then a ≤ c.