Conjunction fallacy

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Conjunction fallacy

The conjunction fallacy is a logical fallacy that occurs when it is assumed that specific conditions are more probable than a single general one.

The most often-cited example of this fallacy originated with Amos Tversky and Daniel Kahneman:[1]

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.

Which is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

85% of those asked chose option 2.[2] However the probability of two events occurring together (in "conjunction") is always less than or equal to the probability of either one occurring alone—formally, for two events A and B this inequality could be written as $\Pr(A \and B) \leq \Pr(A)$, and $\Pr(A \and B) \leq \Pr(B).$

For example, even choosing a very low probability of Linda being a bank teller, say Pr(Linda is a bank teller) = 0.05 and a high probability that she would be a feminist, say Pr(Linda is a feminist) = 0.95, then, assuming independence, Pr(Linda is a bank teller and Linda is a feminist) = 0.05 × 0.95 or 0.0475, lower than Pr(Linda is a bank teller).

Tversky and Kahneman argue that most people get this problem wrong because they use the representativeness heuristic to make this kind of judgment: Option 2 seems more "representative" of Linda based on the description of her, even though it is clearly mathematically less likely.

Many other demonstrations of this error have been studied. In another experiment, for instance, policy experts were asked to rate the probability that the Soviet Union would invade Poland, and the United States would break off diplomatic relations, all in the following year. They rated it on average as having a 4% probability of occurring. Another group of experts was asked to rate the probability simply that the United States would break off relations with the Soviet Union in the following year. They gave it an average probability of only 1%. Researchers argued that a detailed, specific scenario seemed more likely because of the representativeness heuristic, but each added detail would actually make the scenario less and less likely.[3] In this way it could be similar to the misleading vividness or slippery slope fallacies, though it is possible that people underestimate the general possibility of an event occurring when not given a plausible scenario to ponder.

Gigerenzer

Gerd Gigerenzer presents a different take on the conjunction fallacy,[4] and claims that the problem doesn't necessarily lie with the participants, but with the way the question is phrased. First, the words ‘probable’ and ‘and’ can have several meanings. The meaning of probable “what happens frequently”, corresponds to the mathematical probability people are supposed to be tested on. But the meanings of probable “what is plausible”, and “whether there is evidence”, do not (Oxford Dictionary). To explain the participants’ bad performance Gigerenzer takes into account Grice’s (1989) conversational maxims like relevance. In the context of the example people think that the description of Linda is relevant for finding the answer. If the question is rephrased:

There are 100 persons who fit the description above (that is, Linda’s). How many of them are:

Bank tellers? __ of 100

Bank tellers and active in the feminist movement? __ of 100

Whereas previously 85% of participants gave the wrong answer (bank teller and active in the feminist movement) in experiments done with this questioning none of the participants gave a wrong answer (Gigerenzer mentions Hertwig & Gigerenzer, 1999). Readers are sensitive to the different meanings of words, and can understand probabilistic reasoning very well if presented in a more concrete way (like natural frequencies: 1 out of 100 persons instead of 1%). According to Gigerenzer this demonstrates intelligent context-sensitive reasoning.

Notes

1. ^ Tversky & Kahneman (1982, 1983)
2. ^ Many variations of this experiment in wording and framing have been published. When Tversky and Kahneman (1983) changed the first option to "Linda is a bank teller whether or not she is active in the feminist movement" in the same experiment as described a majority of respondents still preferred the second option.
3. ^ Tversky & Kahneman (1983)
4. ^ See Gigerenzer, I Think Therefore I ERR

References

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