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.     

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.     

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.     

Linda versus World Cup: Conjunctive Probabilities in Three-event Fictional and Real-life Predictions

Conjunction errors in probability judgments have been explained in terms of representatives, non-normative combination procedures, and linguistic, conversational, or conceptual misunderstandings. In two studies, a three-event variant of the classical Linda scenario (Tversky and Kahneman, 1983) was contrasted with estimates of Norway’s chances in three coming World Cup soccer matches. Conjunction errors occurred even in the latter, real-life prediction task, but much less frequently than in the fictional Linda case. Magnitude of the conjunction effect was found to be dependent upon type of constituent (fictional versus dispositional), unequal versus equal probabilities of constituent events, prediction of positive versus negative outcomes, and, for real-life predictions only, umber of constitutent events. Fictional probability ratings were close to but lower than representativenss ratings, giving evidence for a representativeness and adjustment-for-uncertainty strategy, whereas probabilities of real-life events were given a causal model interpretation.     

Causal conjunction fallacies: The roles of causal strength and mental resources

In two experiments we tested the prediction derived from Tversky and Kahneman's (1983) work on the causal conjunction fallacy that the strength of the causal connection between constituent events directly affects the magnitude of the causal conjunction fallacy. We also explored whether any effects of perceived causal strength were due to graded output from heuristic Type 1 reasoning processes or the result of analytic Type 2 reasoning processes. As predicted, Experiment 1 demonstrated that fallacy rates were higher for strongly than for weakly related conjunctions. Weakly related conjunctions in turn attracted higher rates of fallacious responding than did unrelated conjunctions. Experiment 2 showed that a concurrent memory load increased rates of fallacious responding for strongly related but not for weakly related conjunctions. We interpret these results as showing that manipulations of the strength of the perceived causal relationship between the conjuncts result in graded output from heuristic reasoning process and that additional mental resources are required to suppress strong heuristic output.     

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.     

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.     

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.     

The conjunction fallacy in profiles of victims of homicide

Using basic probability theory, estimates of the characteristics of the average homicide victim are calculated using the notion of disjoint probability. The assumption of disjoint events (e.g., the victim's race bears no effect on the offender's weapon choice) is then tested empirically using the Uniform Crime Report‐Supplementary Homicide Report. Exploratory results suggest that many demographic and situational characteristics taken together are only slightly more related than chance. Put simply, the average profile of homicide victims portrayed by the media becomes less likely as demographic variables are added. A survey was conducted to test whether individuals conjoined these characteristics, thinking they were more likely to occur together. Consistent with the conjunction fallacy, many participants overestimated the likelihood that certain demographic or situational characteristics will occur together, and some overestimated it to a mathematically impossible degree. These two experiments showcase the difficulty in displaying statistical profiling and how it affects the public's perception of offenders.     

Social Cognition and the So-Called Conjunction Fallacy

The so-called ‘conjunction effect’, in which participants incorrectly assert that an instance from the conjunction of two sets is more probable than an instance from one of the two conjoining sets alone, has been a source of debate as to whether it is a genuine fallacy of individual thinking or not. We argue that reasoning about individuals follows a different process than reasoning about sets. 35 participants took part in 3 tasks: a) one involving blocks of different sizes and colours designed to evoke set-based reasoning, b) one where a particular block was ‘individuated’ by stating that it represented a particular person, and c) the original Tversky and Kahneman (Psychological Review 90(4):293–315, 1983) ‘Linda’ problem. As predicted, set-based reasoning was significantly more prevalent for the blocks task than for the other two tasks. Participants’ reasons for their choices suggest that some individuals correctly use set-based logic in one task and a social reasoning process for the other tasks.     

Understanding conjunction effects in probability judgments: the role of implicit mental models

Subjective probability judgments often violate a normative principle in that the conjunction of two events is judged to be more likely than the probability of either of the two events occurring separately. Most previous explanations of these conjunction effects have assumed that probability judgments depend on some psychological relation (e.g. representativeness) between the constituents mentioned explicitly in the stimulus information. In contrast, the present approach highlights the fundamental role of implicitly inferred information. Participants are assumed to transform the explicit stimulus information into implicit mental models in their attempt to make sense of the experimental task. Probability judgments should then reflect the degree of activation of such a mental model in memory given a set of propositions, rather than the quantitative fit or likelihood of the propositions themselves. Two studies are reported which provide converging evidence for the proposed mental model approach. In the first study, using graded conjunctions of one to five propositions, probability judgments are shown to vary as a function of the activation of a mental model rather than the likelihood of the component events. In a second study, a priming procedure is employed to activate mental models that either fit an event conjunction or do not, leading to an increase or decrease of conjunction effects in probability judgment. Copyright © 1999 John Wiley & Sons, Ltd.     

Information processing and reasoning with premises that are empirically false: Interference,working memory,and processing speed

In this study, we looked at the contributions of individual differences in susceptibility to interference and working memory to logical reasoning with premises that were empirically false (i.e., not necessarily true). A total of 97 university students were given a sentence completion task for which a subset of stimuli was designed to generate inappropriate semantic activation that interfered with the correct response, a measure of working memory capacity, and a series of logical reasoning tasks with premises that were not always true. The results indicate that susceptibility to interference, as measured by the error rate on the relevant subset of the sentence completion task, and working memory independently account for variation in reasoning performance. The participants who made more errors in the relevant portion of the sentence completion task also showed more empirical intrusions in the deductive reasoning task, even when the effects of working memory were partialed out. Working memory capacity was more clearly related to processes involved in generating uncertainty responses to inferences for which there was no certain conclusion. A comparison of the results of this study with studies of children's reasoning suggests that adults are capable of more selective executive processes than are children. An analysis of latency measures on the sentence completion task indicated that high working memory participants who made no errors on the sentence completion task used a strategy that involved slower processing speed, as compared with participants with similar levels of working memory who did make errors. In contrast, low working memory participants who made no errors on the sentence completion task had relatively shorter reaction times than did comparable participants who did make errors.     

主体关联性对贝叶斯推理概率估计的影响

用贝叶斯推理问题为实验材料,探讨了主体关联性对贝叶斯推理概率估计的影响。结果表明,当估计的事件与主体有关时,被试对消极事件概率估计较低,对积极事件概率估计值;当估计的事件与主体无关时,被试对消极事件和积极事件的概率估计无显著差异。反应时分析表明,被试对消极事件的概率估计比对积极事件的概率估计时间显著地长,当消极事件与主体有关时概率估计时间就更长;而对积极事件的概率估计,与主体有关和与主体无关时反应时差异不显著。这表明,被试对消极事件的概率估计(特别是消极事件与己有关时)更为慎重。     

Semantic Coherence and Inconsistency in Estimating Conditional Probabilities

Two studies examined semantic coherence and internal inconsistency fallacies in conditional probability estimation. Problems reflected five distinct relationships between two sets: identical sets, mutually exclusive sets, subsets, overlapping sets, and independent sets (a special case of overlapping sets). Participants estimated P(A), P(B), P(A|B), and P(B|A). Inconsistency occurs when this constellation of estimates does not conform to Bayes' theorem. Semantic coherence occurs when this constellation of estimates is consistent with the depicted relationship among sets. Fuzzy‐trace theory predicts that people have difficulty with overlapping sets and subsets because they require class‐inclusion reasoning. On these problems, people are vulnerable to denominator neglect, the tendency to ignore relevant denominators, making the gist more difficult to discern. Independent sets are simplified by the gist understanding that P(A) provides no information about P(B), and thus, P(A|B) = P(A). The gist for identical sets is that P(A|B) = 1.0, and the gist of mutually exclusive sets is that P(A|B) = 0. In Study 1, identical, mutually exclusive, and independent sets yielded superior performance (in internal inconsistency and semantic coherence) than subsets and overlapping sets. For subsets and overlapping sets, interventions clarifying appropriate denominators generally improved semantic coherence and inconsistency, including teaching people to use Euler diagrams, 2 × 2 tables, or relative frequencies. In Study 2, with problems about breast cancer and BRCA mutations, there was a strong correlation between inconsistency in conditional probability estimation and conjunction fallacies of joint probability estimation, suggesting that similar fallacious reasoning processes produce these errors. Copyright © 2012 John Wiley & Sons, Ltd.     

Expertise and error in diagnostic reasoning

An investigation is presented in which a computer simulation model (DIAGNOSER) is used to develop and test predictions for behavior of subjects in a task of medical diagnosis. The first experiment employed a process-tracing methodology in order to compare hypothesis generation and evaluation behavior of DIAGNOSER with individuals at different levels of expertise (students, trainees, experts). A second experiment performed with only DIAGNOSER identified conditions under which errors in reasoning in the first experiment could be related to interpretation of specific data items. Predictions derived from DIAGNOSER's performance were tested in a third experiment with a new sample of subjects. Data from the three experiments indicated that (1) form of diagnostic reasoning was similar for all subjects trained in medicine and for the simulation model, (2) substance of diagnostic reasoning employed by the simulation model was parable with that of the more expert subjects, and (3) errors in subjects' reasoning were attributable to deficiencies in disease knowledge and the interpretation of specific patient data cues predicted by the simulation model.     

Probabilistic Reasoning in Context: Socio-cultural Differences in Children’s and Adults’ Predictions about the Fulfillment of Prayers and Wishes

ABSTRACT

Children and adults appreciate that physical action is typically the conduit between individuals’ desires and the fulfillment of those desires. However, certain forms of petitionary thought – e.g., wishing and praying – are believed by many people to influence the external world and fulfill desires without direct physical action. We examine whether children’s and adults’ predictions about the occurrence of desired events differs based on: (1) the ways in which desires are expressed (wishing vs. praying), (2) whether desired events are plausible vs. impossible, and (3) participants’ religious backgrounds. Children ages 3- to 11-years (n = 144) and young adults (n = 85) were read scenarios in which a protagonist either wished or prayed for a desired event to occur. Some of the desired events could plausibly happen with ordinary human intervention and others were impossible, even with human intervention. Preschoolers often predicted that desired events would obtain; with increasing age, participants judged that fewer events would obtain. Participants’ predictions varied by the probability of the desired event – across the entire age-range participants predicted that impossible events would obtain less often than plausible events. Thus, participants’ everyday probabilistic reasoning was imported into their reasoning about the fulfillment of supernatural petitions. Children’s and adults’ religious experiences predicted their judgments that events would obtain if they had been prayed for, but not if they had been wished for. Thus, the effects of socio-cultural experience did not globally pervade children’s and adults’ probabilistic reasoning, but were specific to certain contexts about which they reasoned.     

Frequency, probability, and prediction: easy solutions to cognitive illusions?

Many errors in probabilistic judgment have been attributed to people's inability to think in statistical terms when faced with information about a single case. Prior theoretical analyses and empirical results imply that the errors associated with case-specific reasoning may be reduced when people make frequentistic predictions about a set of cases. In studies of three previously identified cognitive biases, we find that frequency-based predictions are different from-but no better than-case-specific judgments of probability. First, in studies of the "planning fallacy, " we compare the accuracy of aggregate frequency and case-specific probability judgments in predictions of students' real-life projects. When aggregate and single-case predictions are collected from different respondents, there is little difference between the two: Both are overly optimistic and show little predictive validity. However, in within-subject comparisons, the aggregate judgments are significantly more conservative than the single-case predictions, though still optimistically biased. Results from studies of overconfidence in general knowledge and base rate neglect in categorical prediction underline a general conclusion. Frequentistic predictions made for sets of events are no more statistically sophisticated, nor more accurate, than predictions made for individual events using subjective probability.     

Noncategorical approaches to feature prediction with uncertain categories

In four experiments, we investigated how people make feature predictions about objects whose category membership is uncertain. Artificial visual categories were presented and remained in view while a novel instance with a known feature, but uncertain category membership was presented. All four experiments showed that feature predictions about the test instance were most often based on feature correlations (referred to as feature conjunction reasoning). Experiment 1 showed that feature conjunction reasoning was generally preferred to category-based induction in a feature prediction task. Experiment 2 showed that people used all available exemplars to make feature conjunction predictions. Experiments 3 and 4 showed that the preference for predictions based on feature conjunction persisted even when category-level information was made more salient and inferences involving a larger number of categories were required. Little evidence of reasoning based on the consideration of multiple categories (e.g., Anderson, (Psychological Review, 98:409–429, 1991)) or the single, most probable category (e.g., Murphy & Ross, (Cognitive Psychology, 27:148–193, 1994)) was found.     

Why versus how often: Causal reasoning and the incidence of judgmental bias

When faced with the task of making a prediction or estimating a likelihood, it is argued that people often reason about the presence or absence and relative strength of possible causal mechanisms for the production of relevant outcomes. In so doing people rely on “causal cues” or properties of an inferential problem which indicate the nature of the particular causal processes which give rise to specific outcomes. It is hypothesized that causal cues, precisely because they focus attention and thought on specific causal mechanisms, can obscure the relevance of mathematical laws of probability and lead to statistically biased judgment. Two experiments were conducted. Their results support the hypothesis, showing that the incidence of the conjunction fallacy and the base rate fallacy depend on task-specific cues for causal reasoning.