Conjunction fallacy
Encyclopedia
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
:
85% of those asked chose option 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
, and 
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. 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.
If the question is rephrased:
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.
The most often-cited example of this fallacy originated with Amos Tversky
Amos Tversky
Amos Nathan Tversky, was a cognitive and mathematical psychologist, a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement...
and Daniel Kahneman
Daniel Kahneman
Daniel Kahneman is an Israeli-American psychologist and Nobel laureate. He is notable for his work on the psychology of judgment and decision-making, behavioral economics and hedonic psychology....
:
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?
- Linda is a bank teller.
- Linda is a bank teller and is active in the feminist movement.
85% of those asked chose option 2. However the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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
, and 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
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
, 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
Representativeness heuristic
The representativeness heuristic is a psychological term describing a phenomenon wherein people judge the probability or frequency of a hypothesis by considering how much the hypothesis resembles available data as opposed to using a Bayesian calculation. While often very useful in everyday life, it...
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
Soviet Union
The Soviet Union , officially the Union of Soviet Socialist Republics , was a constitutionally socialist state that existed in Eurasia between 1922 and 1991....
would invade Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...
, and the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
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
Representativeness heuristic
The representativeness heuristic is a psychological term describing a phenomenon wherein people judge the probability or frequency of a hypothesis by considering how much the hypothesis resembles available data as opposed to using a Bayesian calculation. While often very useful in everyday life, it...
, but each added detail would actually make the scenario less and less likely. In this way it could be similar to the misleading vividness
Misleading vividness
Misleading vividness is a term that can be applied to anecdotal evidence describing an occurrence, even if it is an exceptional occurrence, with sufficient detail to permit hasty generalizations about the occurrence...
or slippery slope
Slippery slope
In debate or rhetoric, a slippery slope is a classic form of argument, arguably an informal fallacy...
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, 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.