Psychological Review VOLUME 90 NUMBER 4 OCTOBER 1983 Extensional Versus Intuitive Reasoning: The Conjunction Fallacy in Probability Judgment Amos Tversky Daniel Kahneman Stanford University University of British Columbia, Vancouver, British Columbia, Canada Perhaps the simplest and the most basic qualitative law of probability is the con- junction rule: The probability of a conjunction, P(A&B), cannot exceed the prob- abilities of its constituents, P(A) and .P(B), because the extension (or the possibility set) of the conjunction is included in the extension of its constituents. Judgments under uncertainty, however, are often mediated by intuitive heuristics that are not bound by the conjunction rule. A conjunction can be more representative than one of its constituents, and instances of a specific category can be easier to imagine or to retrieve than instances of a more inclusive category. The representativeness and availability heuristics therefore can make a conjunction appear more probable than one of its constituents. This phenomenon is demonstrated in a variety of contexts including estimation of word frequency, personality judgment, medical prognosis, decision under risk, suspicion of criminal acts, and political forecasting. Systematic violations of the conjunction rule are observed in judgments of lay people and of experts in both between-subjects and within-subjects comparisons. Alternative interpretations of the conjunction fallacy are discussed and attempts to combat it are explored. Uncertainty is an unavoidable aspect of the the last decade (see, e.g., Einhorn & Hogarth, human condition. Many significant choices 1981; Kahneman, Slovic, & Tversky, 1982; must be based on beliefs about the likelihood Nisbett & Ross, 1980). Much of this research of such uncertain events as the guilt of a de- has compared intuitive inferences and prob- fendant, the result of an election, the future ability judgments to the rules of statistics and value of the dollar, the outcome of a medical the laws of-probability. The student of judg- operation, or the response of a friend. Because ment uses the probability calculus as a stan- we normally do not have adequate formal dard of comparison much as a student of per- models for computing the probabilities of such ception might compare the perceived sizes of events, intuitive judgment is often the only objects to their physical sizes. Unlike the cor- practical method for assessing uncertainty. rect size of objects, however, the "correct" The question of how lay people and experts probability of events is not easily defined. Be- evaluate the probabilities of uncertain events cause individuals who have different knowl- has attracted considerable research interest in edge or who hold different beliefs must be al- lowed to assign different probabilities to the This research was supported by Grant NR197-058 from same event> no sing!e value can be correct for the U.S. Office of Naval Research. We are grateful to friends all people. Furthermore, a correct probability and colleagues, too numerous to list by name, for their cannot always be determined even for a single useful comments and suggestions on an earlier draft of person Outside the domain of random sam- 'Xuets for reprints should be sent to Amos Tversky, PlinS> Probability theory does not determine Department of Psychology, Jordan Hall, Building 420, the probabilities of uncertain events—it merely Stanford University, Stanford, California 94305. imposes constraints on the relations among Copyright 1983 by the American Psychological Association, Inc. 293 294 AMOS TVERSKY AND DANIEL KAHNEMAN them. For example, if A is more probable than sessments that are routinely carried out as part B, then the complement of A must be less of the perception of events and the compre- probable than the complement of B. hension of messages. Such natural assessments The laws of probability derive from exten- include computations of similarity and rep- sional considerations. A probability measure resentativeness, attributions of causality, and is defined on a family of events and each event evaluations of the availability of associations is construed as a set of possibilities, such as and exemplars. These assessments, we propose, the three ways of getting a 10 on a throw of are performed even in the absence of a specific a pair of dice. The probability of an event task set, although their results are used to meet equals the sum of the probabilities of its dis- task demands as they arise. For example, the joint outcomes. Probability theory has tradi- mere mention of "horror movies" activates tionally been used to analyze repetitive chance instances of horror movies and evokes an as- processes, but the theory has also been applied sessment of their availability. Similarly, the to essentially unique events where probability statement that Woody Allen's aunt had hoped is not reducible to the relative frequency of that he would be a dentist elicits a comparison "favorable" outcomes. The probability that the of the character to the stereotype and an as- man who sits next to you on the plane is un- sessment of representativeness. It is presum- married equals the probability that he is a ably the mismatch between Woody Allen's bachelor plus the probability that he is either personality and our stereotype of a dentist that divorced or widowed. Additivity applies even makes the thought mildly amusing. Although when probability does not have a frequentistic these assessments are not tied to the estimation interpretation and when the elementary events of frequency or probability, they are likely to are not equiprobable. play a dominant role when such judgments The simplest and most fundamental qual- are required. The availability of horror movies itative law of probability is the extension rule: may be used to answer the question, "What If the extension of A includes the extension proportion of the movies produced last year of B (i.e., A D B) then P(A) > P(B). Because were horror movies?", and representativeness the set of possibilities associated with a con- may control the judgment that a particular junction A&B is included in the set of pos- boy is more likely to be an actor than a dentist. sibilities associated with B, the same principle The term judgmental heuristic refers to a can also be expressed by the conjunction rule strategy-whether deliberate or not—that relies /•(A&B) < P(B): A conjunction cannot be on a natural assessment to produce an esti- more probable than one of its constituents. mation or a prediction. One of the manifes- This rule holds regardless of whether A and tations of a heuristic is the relative neglect of B are independent and is valid for any prob- other considerations. For example, the resem- ability assignment on the same sample space. blance of a child to various professional ste- Furthermore, it applies not only to the stan- reotypes may be given too much weight in dard probability calculus but also to nonstan- predicting future vocational choice, at the ex- dard models such as upper and lower prob- pense of other pertinent data such as the base- ability (Dempster, 1967;Suppes, 1975), belief rate frequencies of occupations. Hence, the function (Shafer, 1976), Baconian probability use of judgmental heuristics gives rise to pre- (Cohen, 1977), rational belief (Kyburg, in dictable biases. Natural assessments can affect press), and possibility theory (Zadeh, 1978). judgments in other ways, for which the term In contrast to formal theories of belief, in- heuristic is less apt. First, people sometimes tuitive judgments of probability are generally misinterpret their task and fail to distinguish not extensional. People do not normally an- the required judgment from the natural as- alyze daily events into exhaustive lists of pos- sessment that the problem evokes. Second, the sibilities or evaluate compound probabilities natural assessment may act as an anchor to by aggregating elementary ones. Instead, they which the required judgment is assimiliated, commonly use a limited number of heuristics, even when the judge does not intend to use such as representativeness and availability the one to estimate the other. (Kahneman et al., 1982). Our conception of Previous discussions of errors of judgment judgmental heuristics is based on natural as- have focused on deliberate strategies and on PROBABILITY JUDGMENT 295 misinterpretations of tasks. The present treat- comparison of words of the form / y ment calls special attention to the processes with words of the form / _; the me- of anchoring and assimiliation, which are often dian estimates were 8.8 and 4.4, respectively. neither deliberate nor conscious. An example This example illustrates the structure of the from perception may be instructive: If two studies reported in this article. We constructed objects in a picture of a three-dimensional problems in which a reduction of extension scene have the same picture size, the one that was associated with an increase in availability appears more distant is not only seen as or representativeness, and we tested the con- "really" larger but also as larger in the picture. junction rule in judgments of frequency or The natural computation of real size evidently probability. In the next section we discuss the influences the (less natural) judgment of pic- representativeness heuristic and contrast it ture size, although observers are unlikely to with the conjunction rule in the context of confuse the two values or to use the former person perception. The third section describes to estimate the latter. conjunction fallacies in medical prognoses, The natural assessments of representative- sports forecasting, and choice among bets. In ness and availability do not conform to the the fourth section we investigate probability extensional logic of probability theory. In par- judgments for conjunctions of causes and ef- ticular, a conjunction can be more represen- fects and describe conjunction errors in sce- tative than one of its constituents, and in- narios of future events.