How probability theory explains the conjunction fallacy

The conjunction fallacy occurs when people judge a conjunctive statement B‐and‐A to be more probable than a constituent B, in contrast to the law of probability that P(B ∧ A) cannot exceed P(B) or P(A). Researchers see this fallacy as demonstrating that people do not follow probability theory when judging conjunctive probability. This paper shows that the conjunction fallacy can be explained by the standard probability theory equation for conjunction if we assume random variation in the constituent probabilities used in that equation. The mathematical structure of this equation is such that random variation will be most likely to produce the fallacy when one constituent has high probability and the other low, when there is positive conditional support between the constituents, when there are two rather than three constituents, and when people rank probabilities rather than give numerical estimates. The conjunction fallacy has been found to occur most frequently in exactly these situations. Copyright © 2008 John Wiley & Sons, Ltd.     

Walter the banker: the conjunction fallacy reconsidered

In a famous experiment by Tversky and Kahneman (Psychol Rev 90:293–315, 1983), featuring Linda the bank teller, the participants assign a higher probability to a conjunction of propositions than to one of the conjuncts, thereby seemingly committing a probabilistic fallacy. In this paper, we discuss a slightly different example featuring someone named Walter, who also happens to work at a bank, and argue that, in this example, it is rational to assign a higher probability to the conjunction of suitably chosen propositions than to one of the conjuncts. By pointing out the similarities between Tversky and Kahneman’s experiment and our example, we argue that the participants in the experiment may assign probabilities to the propositions in question in such a way that it is also rational for them to give the conjunction a higher probability than one of the conjuncts.     

Paranormal belief and errors of probabilistic reasoning: The role of constituent conditional relatedness in believers' susceptibility to the conjunction fallacy

The present study examines the extent to which stronger belief in either extrasensory perception, psychokinesis or life-after-death is associated with a proneness to making conjunction errors (CEs). One hundred and sixty members of the UK public read eight hypothetical scenarios and for each estimated the likelihood that two constituent events alone plus their conjunction would occur. The impact of paranormal belief plus constituents’ conditional relatedness type, estimates of the subjectively less likely and more likely constituents plus relevant interaction terms tested via three Generalized Linear Mixed Models. General qualification levels were controlled for. As expected, stronger PK beliefs and depiction of a positively conditionally related (verses conditionally unrelated) constituent pairs predicted higher CE generation. ESP and LAD beliefs had no impact with, surprisingly, higher estimates of the less likely constituent predicting fewer - not more - CEs. Theoretical implications, methodological issues and ideas for future research are discussed.     

Religious versus reflective priming and susceptibility to the conjunction fallacy

The effect of religious priming has been studied in relation to a number of variables, most extensively with prosocial behavior. The effects of priming on cognitive domains, however, are relatively understudied. The present study examined the effects of religious priming, compared with reflective and neutral priming, on the conjunction fallacy. Participants were randomly assigned to 1 of the 3 priming conditions. Priming was presented through the scrambled sentence task in which participants were required to rearrange words of a religious (e.g., pray), reflective (e.g., reason), or neutral (e.g., paper) content. The conjunction fallacy was measured by a task containing 1 problem. Results indicated that those undergoing the religious prime were significantly more likely to commit the conjunction fallacy compared with those in the reflective priming group. Situations in which reasoning is integral may benefit from knowing the immediate effects of religious versus reflective stimuli in the environment.     

The dependence of the conjunction fallacy on subtle linguistic factors

Summary The three experiments reported in this article are concerned with moderating conditions of the so-called conjunction fallacy. Although the conjunction of two events (A&B) is necessarily less probable than one event alone, intelligent people's judgments sometimes violate this logical principle when it is easier to think about or imagine the conjunction A&B than the component event A. It was hypothesized that the fallacy might be reduced either by a priming treatment (directing the participants' attention to the logical rule prior to the conjunctions tasks) or by a linguistic manipulation (replacing the ambiguous term probability by the term frequency. Experiment 1 shows that the bias is drastically reduced by the subtle linguistic manipulation and suggests that the fallacy is partly due to a common misunderstanding of the probability concept. The lack of a priming effect seems to imply that cueing or activation of logical schemata is not a critical factor. Experiment 2 replicates the linguistic influence under slightly modified task conditions, and Experiment 3 corroborates the ineffectiveness of the priming factor using a stronger priming treatment.     

Suppressing natural heuristics by formal instruction: The case of the conjunction fallacy

A basic principle of probability is the conjunction rule, p(B) p(A&B). People violate this rule often, particularly when judgments of probability are based on intensional heuristics such as representativeness and availability. Though other probabilistic rules are obeyed with increasing frequency as people's levels of mathematical talent and training increase, the conjunction rule generally does not show such a correlation. We argue that this recalcitrance is not due to inescapable “natural assessments”; rather, it stems from the absence of generally useful problem-solving designs that bring extensional principles to bear on this class of problem. We predict that when helpful extensional strategies are made available, they should compete well with intensional heuristics. Two experiments were conducted, using as subjects adult women with little mathematical background. In Experiment 1, brief training on concepts of algebra of sets, with examples of their use in solving problems, reduced conjunction-rule violations substantially, compared with a control group. Evidence from similarity judgments suggested that use of the representativeness heuristic was reduced by the training. Experiment 2 confirmed these training effects and also tested the hypothesis that conjunction-rule violations are due to misunderstanding of “B” as “B and not A.” Changes in detailed wording of the propositions to be ranked produced substantial effects on judgment, but the pattern of these effects supported the hypothesis that, for the type of problem used here, most conjunction errors are due to use of representativeness or availability. We conclude that such intensional heuristics can be suppressed when alternative strategies are taught.     

On the conjunction fallacy and the meaning of and, yet again: a reply to Hertwig, Benz, and Krauss (2008)

In this paper we question the theoretical tenability of Hertwig, Benz, and Krauss's (2008) (HBK) argument that responses commonly taken as manifestations of the conjunction fallacy should be instead considered as reflecting "reasonable pragmatic and semantic inferences" because the meaning of and does not always coincide with that of the logical operator ∧. We also question the relevance of the experimental evidence that HBK provide in support of their argument as well as their account of the pertinent literature. Finally, we report two novel experiments in which we employed HBK's procedure to control for the interpretation of and. The results obtained overtly contradict HBK's data and claims. We conclude with a discussion on the alleged feebleness of the conjunction fallacy, and suggest directions that future research on this topic might pursue.     

Some remarks on the Rationality of Induction

This paper begins with a rigorous critique of David Stove's recent bookThe Rationality of Induction. In it, Stove produced four different proofs to refute Hume's sceptical thesis about induction. I show that Stove's attempts to vindicate induction are unsuccessful. Three of his proofs refute theses that are not the sceptical thesis about induction at all. Stove's fourth proof, which uses the sampling principle to justify one particular inductive inference, makes crucial use of an unstated assumption regarding randomness. Once this assumption is made explicit, Hume's thesis once more survives.The refutation of Stove's fourth proof leads to some observations which relate Goodman's grue paradox with randomness of a sample. I formulate a generalized version of Goodman's grue paradox, and argue that whenever a sample, no matter how large, is drawn from a predetermined smaller interval of a population that is distributed over a larger interval, any conclusion drawn about the characteristics of the population based on the observed characteristics of the sample is fatally vulnerable to the generalized grue paradox.Finally, I argue that the problem of justification of induction can be addressed successfully only from a cognitive point of view, but not from a metaphysical one. That is, we may ask whether an inductive inference is justified or not within the theories or cognitive structures of a subject, but not outside them. With this realization, induction is seen as a cognitive process, not unlike vision, that is useful at times, and yet has its own illusions that may make it a serious obstacle to cognition at other times.     

The illusion of replacement in research into the development of thinking biases: the case of the conjunction fallacy

The conjunction fallacy is a violation of a very basic rule of probability. Interestingly, although committing the fallacy seems irrational, adults are no less susceptible to the fallacy than young children. In Experiment 1, by employing tasks where the conjunctive response option involved two non-representative items, we found a large reduction in fallacy rates as compared to traditional conjunction fallacy problems. Nevertheless, fallacy rates remained relatively high in both adolescents and adults, although adults showed more consistency in their normative responses. In Experiment 2, we demonstrated that children’s relatively good performance on the task was not the consequence of their missing knowledge of social stereotypes. Additionally, children were more strongly affected by explicitly presented frequency information than adults. Indeed, adults only took frequency information into account when frequencies were made relevant by a training in probabilistic reasoning. Overall, the results suggest that whereas the potential for normative reasoning increases with development, this potential is often overshadowed by a pervasive tendency in adolescence and adulthood to rely on contextual information, knowledge, and beliefs, even when conflicting information is available. By contrast, children are more strongly influenced by explicitly presented information than relevant knowledge cued by the tasks.     

Paranormal belief and the conjunction fallacy: Controlling for temporal relatedness and potential surprise differentials in component events

Recent research suggests paranormal believers are especially prone to the ‘conjunction fallacy’. The current study extends this work by presenting believers and non‐believers with eight paranormal plus eight non‐paranormal scenarios. Participants were given either a paranormal or virtually identical non‐paranormal version of each scenario. Of these, half incorporated component events which were (virtually) co‐occurring with half including components which were temporally disjointed. Analysis of Covariance (ANCOVA; controlling for gender and maths/stats/psychology qualifications) found believers made more conjunction errors than non‐believers. Neither event type (paranormal vs. non‐paranormal) nor components' temporal relationship (co‐occurring vs. disjointed) had a significant effect on conjunction biases. Believers' tendency to produce larger conjunctive estimates was unrelated to group differences in component probability estimates (surprise values) and further, could not be attributed to group differences in the perceived functional relationship between component and conjunctive events. Possible explanations are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Arithmetic and geometry: Some remarks on the concept of complementarity

This paper explores the classical idea of complementarity in mathematics concerning the relationship of intuition and axiomatic proof. Section I illustrates the basic concepts of the paper, while Section II presents opposing accounts of intuitionist and axiomatic approaches to mathematics. Section III analyzes one of Einstein's lecture on the topic and Section IV examines an application of the issues in mathematics and science education. Section V discusses the idea of complementarity by examining one of Zeno's paradoxes. This is followed by presenting a few more programmatic suggestions and a brief summary.     

Validity generalization: Some critical remarks on the Schmidt-Hunter procedure*

This article presents some critical comments on the validity generalization procedure which has been presented by Schmidt, Hunter and others. They have put forward a method for testing the hypothesis that the variance in validity coefficients across situations for job-test combinations is due to what they consider to be statistical artifacts. The Schmidt-Hunter approach is criticized on the following points: the compilation of validity data, the use of criterion measures, and the test of the hypothesis of no situational specificity. Further, the relation between the concepts ‘situational specificity’ and ‘validity generalization’ is considered. In addition, it is noted that Schmidt, Hunter and others have defined the concept ‘situation’ in a different way than classical writers. It is concluded that the Schmidt-Hunter approach to validity generalization shows fundamental shortcomings. As a consequence their far-reaching conclusions for the practice of personnel selection should be considered premature.     

Some remarks on failure to meet assumptions in discriminant analyses

The linear discriminant function and the generalized distance function, two special cases of discriminant technique, require multivariate normality and homogeneous variance-covariance matrices, and hence utilize only mean differences among groups. The more general methods can also utilize differences in variances and/or covariances. Tables are given showing the discriminatory value of differences in means, variances, and intercorrelations, taken singly. Equations which utilize all such differences are given for the normal multivariate distribution.The Psychological Corporation     

Testing boundary conditions for the conjunction fallacy: effects of response mode, conceptual focus, and problem type

Two experiments used within-subject designs to examine how conjunction errors depend on the use of (1) choice versus estimation tasks, (2) probability versus frequency language, and (3) conjunctions of two likely events versus conjunctions of likely and unlikely events. All problems included a three-option format verified to minimize misinterpretation of the base event. In both experiments, conjunction errors were reduced when likely events were conjoined. Conjunction errors were also reduced for estimations compared with choices, with this reduction greater for likely conjuncts, an interaction effect. Shifting conceptual focus from probabilities to frequencies did not affect conjunction error rates. Analyses of numerical estimates for a subset of the problems provided support for the use of three general models by participants for generating estimates. Strikingly, the order in which the two tasks were carried out did not affect the pattern of results, supporting the idea that the mode of responding strongly determines the mode of thinking about conjunctions and hence the occurrence of the conjunction fallacy. These findings were evaluated in terms of implications for rationality of human judgment and reasoning.     

Replication is not coincidence: Reply to Iverson, Lee, and Wagenmakers (2009)

Iverson, Lee, and Wagenmakers (2009) claimed that Killeen’s (2005) statistic prep overestimates the “true probability of replication.” We show that Iverson et al. confused the probability of replication of an observed direction of effect with a probability of coincidence—the probability that two future experiments will return the same sign. The theoretical analysis is punctuated with a simulation of the predictions of prep for a realistic random effects world of representative parameters, when those are unknown a priori. We emphasize throughout that prep is intended to evaluate the probability of a replication outcome after observations, not to estimate a parameter. Hence, the usual conventional criteria (unbiasedness, minimum variance estimator) for judging estimators are not appropriate for probabilities such as p and p_rep.