Explaining High Conjunction Fallacy Rates: The Probability Theory Plus Noise Account

The conjunction fallacy occurs when people judge the conjunctive probability P(A ∧ B) to be greater than a constituent probability P(A), contrary to the norms of probability theory. This fallacy is a reliable, consistent and systematic part of people's probability judgements, attested in many studies over at least 40 years. For some events, these fallacies occur very frequently in people's judgements (at rates of 80% or more), while for other events, the fallacies are very rare (occurring at rates of 10% or less). This wide range of fallacy rates presents a challenge for current theories of the conjunction fallacy. We show how this wide range of observed fallacy rates can be explained by a simple model where people reason according to probability theory but are subject to random noise in the reasoning process. Copyright © 2016 John Wiley & Sons, Ltd.     

Fallacies in probability judgments for conjunctions and disjunctions of everyday events

The conjunction fallacy occurs when people judge a conjunction B‐and‐A as more probable than a constituent B, contrary to probability theory's ‘conjunction rule’ that a conjunction cannot be more probable than either constituent. Many studies have demonstrated this fallacy in people's reasoning about various experimental materials. Gigerenzer objects that from a ‘frequentist’ standpoint probability theory is not valid for these materials, and so failure to follow the conjunction rule is not a fallacy. This paper describes three experiments showing that the conjunction fallacy occurs as consistently for conjunctions where frequentist probability theory is valid (conjunctions of everyday weather events) as for other conjunctions. These experiments also demonstrate a reliable correlation between the occurrence of the conjunction fallacy and the disjunction fallacy (which arises when a disjunction B‐or‐A is judged less probable than a constituent B). This supports a probability theory + random variation account of probabilistic reasoning. Copyright © 2008 John Wiley & Sons, Ltd.     

Exploring the conjunction fallacy within a category learning framework

The literature presents two major theories on the cause of the conjunction fallacy. The first attributes the conjunction fallacy to the representativeness heuristic. The second suggests that the conjunction fallacy is caused by people combining p(A) and p(B) into p(A&B) in an inappropriate manner. These two theories were contrasted in two category‐learning experiments. As predicted by the latter theory, data showed that participants that could assess p(A&B) directly made fewer conjunction fallacies than participants who had to compute p(A) and p(B) separately and then combine them into p(A&B). Least conjunction fallacies were observed in the cases where the representativeness heuristic was applicable. Overall, data showed that an inability to appropriately combine probabilities is one of the key cognitive mechanisms behind the conjunction fallacy. Copyright © 2008 John Wiley & Sons, Ltd.     

Naive Probability: Model‐Based Estimates of Unique Events

We describe a dual‐process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non‐numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non‐numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P(A), P(B), P(A and B), disjunctions: P(A), P(B), P(A or B or both), and conditional probabilities P(A), P(B), P(B|A). They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning.     

Paranormal belief and susceptibility to the conjunction fallacy

Numerous studies have shown paranormal believers misperceive randomness and are poor at judging probability. Despite the obvious relevance to many types of alleged paranormal phenomena, no one has examined whether believers are more susceptible to the ‘conjunction fallacy’; that is to misperceiving co‐occurring (conjunct) events as being more likely than singular (constituent) events alone. The present study examines believer vs. non‐believer differences in conjunction errors for both paranormal and non‐paranormal events presented as either a probability or a frequency estimation task. As expected, believers made more conjunction errors than non‐believers. This was true for both event types, with both groups making fewer errors for paranormal than for non‐paranormal events. Surprisingly, the response format (probability vs. frequency) had little impact. Results are discussed in relation to paranormal believers' susceptibility to the conjunction fallacy and more generally, to their propensity for probabilistic reasoning biases. Copyright © 2008 John Wiley & Sons, Ltd.     

Linda问题：“齐当别”抉择模型的解释

大量有关人类归因判断的研究表明,人类经常违反理性概率公理。Tversky和Kahneman（1983）使用Linda问题等特定场景的研究发现,人们系统性地表现出违反理性推断标准,判断合取事件发生概率大于其组成事件发生概率,称之为合取谬误,并用人们使用代表性启发式判断概率来解释该现象产生的原因。然而使用启发式观点对合取谬误现象进行解释过于模糊不清。该文首先介绍了合取谬误现象及其解释模型,然后应用Li（1994,2004）提出的不确定情形下决策理论——“齐当别”抉择模型对Linda问题中合取谬误产生的原因进行了新的解释     

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.     

Reasoning about conjunctive probabilistic concepts in childhood.

While adults are known to exhibit biases when making conjunctive probability judgments, little is known about childhood competencies in this area. Participants (aged between four and five years, eight and ten years, and a group of young adults) attempted to select the more likely of two events, a single event, and a conjunctive event containing, as one of its components, the single event. The problems were such that the objective probabilities of the component events were potentially available. Children in both age groups were generally successful when the single event was likely. However, when it was unlikely, a majority of children rejected it, choosing the conjunctive event instead, thereby committing the conjunction fallacy. A substantial minority of adults also committed the fallacy under equivalent conditions. It is concluded that under certain conditions children are capable of normative conjunctive judgments but that the mechanisms underpinning this capacity remain to be fully understood.     

Conditional Probabilities, Potential Surprise, and the Conjunction Fallacy

Tversky and Kahneman (1983) found that a relationship of positive conditional dependence between the components of a conjunction of two events increases the prevalence of the conjunction fallacy. Consistent with this finding, the results of two experiments reveal that dependence leads to higher estimates for the conjunctive probability and a higher incidence of the fallacy. However, contrary to the theoretical account proposed by Tversky and Kahneman, the actual magnitude of the conditional relationship does not directly affect the extent of the fallacy; all that is necessary is for a positive conditional relationship to exist. The pattern of results obtained can be accounted for in terms of Shackle's (1969) 'potential surprise' theory of subjective probability. Surprise theory predicts that the impact of the conditional event will be at its maximum where the relationship is a negative one. Tversky and Kahneman's model, on the other hand, predicts the maximum effect when the relationship is positive. In all 12 scenarios tested, multiple regression analysis revealed that the standardized beta weight associated with the conditional event was greater when the relationship was a negative one. Thus the outcome was supportive of the surprise model rather than Tversky and Kahneman's account.     

Representativeness and conjoint probability

People commonly violate a basic rule of probability, judging a conjunction of events to be more probable than at least 1 of its component events. Many manifestations of this conjunction fallacy have been ascribed to people's reliance on the representativeness heuristic for judging probability. Some conjunction fallacies, however, have been ascribed to the incorrect rules people use to combine probabilities. In 2 experiments, representativeness was pitted against probability combination to determine the contributions of each to the fallacy. Even for exemplar representativeness problems, the fallacy stemmed primarily from the application of incorrect combination rules. Representativeness seemed to be involved only insofar as it influenced the probabilities of a conjunction's component events. Implications of these findings are discussed for the representativeness account of judgmental errors and the relation between similarity and probability.     

Frequency illusions and other fallacies

[Cosmides and Tooby, 1996] increased performance using a frequency rather than probability frame on a problem known to elicit base-rate neglect. Analogously, [Gigerenzer, 1994] claimed that the conjunction fallacy disappears when formulated in terms of frequency rather than the more usual single-event probability. These authors conclude that a module or algorithm of mind exists that is able to compute with frequencies but not probabilities. The studies reported here found that base-rate neglect could also be reduced using a clearly stated single-event probability frame and by using a diagram that clarified the critical nested-set relations of the problem; that the frequency advantage could be eliminated in the conjunction fallacy by separating the critical statements so that their nested relation was opaque; and that the large effect of frequency framing on the two problems studied is not stable. Facilitation via frequency is a result of clarifying the probabilistic interpretation of the problem and inducing a representation in terms of instances, a form that makes the nested-set relations amongst the problem components transparent.     

The conjunction fallacy?

Tversky and Kahneman (1983) showed that when subjects are asked to rate the likelihood of several alternatives, including single and joint events, they often make a "conjunction fallacy." That is, they rate the conjunction of two events as being more likely than one of the constituent events. This, they claim, is a fallacy, since the conjunction of two events can never be more probable than either of the component events. In addition, they found that prior training in probability theory does not decrease the likelihood of making this fallacy. We argue that in some contexts, an alternative that contains the conjunction of two events can be more probable than an alternative that contains only one of the conjunction's constituent events. We carried out four experiments in which we manipulated this context. The frequency of making a conjunction fallacy was affected by the manipulation of context. Furthermore, when the context was clearly specified, prior training in statistics influenced the ratings.     

Age-trend-related differences in tasks involving conjunctive probabilistic reasoning

Conjunctive probabilistic reasoning has been studied at different ages to ascertain whether the conjunction fallacy is due to a task demand misinterpretation. Such a misinterpretation might occur because a task that requires a comparison between a superordinate class A and a subordinate class A&B is mistakenly interpreted as requiring a comparison between the two complementary subordinate classes of A (i.e., A&B and A¬B). Children (7- and 10-year-olds) and adults were required to make conjunctive probability judgments about problems for which explicit objective probabilities were provided. The total number of A items was kept constant and the frequencies of the A&B and of the A¬B items varied across problems. When the number of A&B items was smaller than the number of A¬B items, the frequency of congruent responses increased with age. When the number of A&B items was greater or equal to that of the A¬B items, the frequency of correct answers decreased. (PsycINFO Database Record (c) 2008 APA, all rights reserved).     

Probabilistic reasoning in prediction and diagnosis: Effects of problem type,response mode,and individual differences

In prediction, subset relations require that the probability of conjoined events is never higher than that of constituent events. However, people's judgments regularly violate this principle, producing conjunction errors. In diagnosis, the probability of a hypothesis normatively is often higher for conjoined cues. An online survey used a within‐subjects design to explore the degree to which participants (n = 347) differentiated diagnosis and prediction using matched scenarios and both choice and estimation responses. Conjunctions were judged more probable than a constituent in diagnosis (76%) more often than prediction (64%) and in choice (84%) more often than direct estimation (57%), with no interaction of type of task and response mode. Correlation, regression, and path analyses were used to determine the relationships among individual difference variables and the diagnosis and prediction tasks. Among the correlation findings was that time spent on the task predicted higher conjunction probabilities in diagnosis but not prediction and that class inclusion errors predicted increased conjunction errors in choice but not estimation. Need for cognition and numeracy were only minimally related to reasoning about conjunctions. Results are consistent with the idea that people may misapply diagnostic reasoning to the prediction task and consequently commit the conjunction error. Copyright © 2010 John Wiley & Sons, Ltd.     

认知需要和警告类型对合取谬误的影响

合取谬误是一种常见的判断偏差,它指的是在不确定条件下,个体评估合取事件及其简单事件发生的概率时,对合取规则系统性偏离的一种现象.实验1 就认知需要类型对合取谬误的影响进行探讨,结果发现高认知需要的被试较不易表现出双重合取谬误和单合取谬误.实验2 探讨了警告类型对合取谬误的影响,结果发现无警告时个体最易表现出单合取谬误,其次是间接警告,最后是直接警告;此外,认知需要与警告类型的交互作用显著,高认知需要的被试在直接警告和间接警告时更少表现出双重合取谬误,在直接警告时更少表现出单合取谬误.     

Is the conjunction fallacy tied to probabilistic confirmation?

Crupi et al. (2008) offer a confirmation-theoretic, Bayesian account of the conjunction fallacy—an error in reasoning that occurs when subjects judge that Pr(h ₁ & h ₂|e) > Pr(h ₁|e). They introduce three formal conditions that are satisfied by classical conjunction fallacy cases, and they show that these same conditions imply that h ₁ & h ₂ is confirmed by e to a greater extent than is h ₁ alone. Consequently, they suggest that people are tracking this confirmation relation when they commit conjunction fallacies. I offer three experiments testing the merits of Crupi et al.’s account specifically and confirmation-theoretic accounts of the conjunction fallacy more generally. The results of Experiment 1 show that, although Crupi et al.’s conditions do seem to be causally linked to the conjunction fallacy, they are not necessary for it; there exist cases that do not meet their three conditions in which subjects still tend to commit the fallacy. The results of Experiments 2 and 3 show that Crupi et al.’s conditions, and those offered by other confirmation-theoretic accounts of the fallacy, are not sufficient for the fallacy either; there exist cases that meet all three of CFT’s conditions in which subjects do not tend to commit the fallacy. Additionally, these latter experiments show that such confirmation-theoretic conditions are at best only weakly causally relevant to the presence of the conjunction fallacy. Given these findings, CFT’s account specifically, and any general confirmation-theoretic account more broadly, falls short of offering a satisfying explanation of the presence of the conjunction fallacy.     

Paranormal Believers' Susceptibility to Confirmatory Versus Disconfirmatory Conjunctions

This study examines paranormal believers' susceptibility to the conjunction fallacy for confirmatory versus non‐confirmatory conjunctive events. Members of the UK public (N = 207) read 16 hypothetical vignettes before judging the likelihood that each constituent and their conjunction would (co) occur. Event type (paranormal versus non‐paranormal), outcome type (confirming versus disconfirming) and level of paranormal belief (in either extrasensory perception, psychokinesis or life after death)—plus relevant interaction terms—were entered into a linear mixed model analysis. As hypothesised, paranormal belief was associated with more conjunction errors regardless of event type with, in general, more errors made for confirmatory over disconfirmatory conjunctions. These trends existed for extrasensory perception and psychokinesis believers with those for life after death believers approaching significance. Consistent with Crupi and Tentori's Confirmation–Theoretical Framework, current findings suggest that paranormal believers are prone to a generic and confirmatory conjunction fallacy. Theoretical implications, methodological limitations and future research ideas are discussed. Copyright © 2016 John Wiley & Sons, Ltd.     

The ‘conjunction fallacy’ revisited: how intelligent inferences look like reasoning errors

Findings in recent research on the ‘conjunction fallacy’ have been taken as evidence that our minds are not designed to work by the rules of probability. This conclusion springs from the idea that norms should be content‐blind—in the present case, the assumption that sound reasoning requires following the conjunction rule of probability theory. But content‐blind norms overlook some of the intelligent ways in which humans deal with uncertainty, for instance, when drawing semantic and pragmatic inferences. In a series of studies, we first show that people infer nonmathematical meanings of the polysemous term ‘probability’ in the classic Linda conjunction problem. We then demonstrate that one can design contexts in which people infer mathematical meanings of the term and are therefore more likely to conform to the conjunction rule. Finally, we report evidence that the term ‘frequency’ narrows the spectrum of possible interpretations of ‘probability’ down to its mathematical meanings, and that this fact—rather than the presence or absence of ‘extensional cues’—accounts for the low proportion of violations of the conjunction rule when people are asked for frequency judgments. We conclude that a failure to recognize the human capacity for semantic and pragmatic inference can lead rational responses to be misclassified as fallacies. Copyright © 1999 John Wiley & Sons, Ltd.