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.     

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.     

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.     

概率判断中的合取谬误

概率判断中的合取谬误是指违反事件发生概率的合取规则而认为包含多个独立事件的复合事件的发生可能性大于其中某些事件的发生可能性的一种概率判断偏差现象。合取谬误的界定存在一定争议, 相关的解释机制有因果模型理论、确认理论、惊奇理论等, 影响合取谬误的因素有频率效应、训练效应以及个体差异等等。未来研究应联系逆转合取谬误的心理机制来完善已有的理论, 同时注意应用研究以及其非理性的探讨。     

On the nature of the conjunction fallacy

In a seminal work, Tversky and Kahneman showed that in some contexts people tend to believe that a conjunction of events (e.g., Linda is a bank teller and is active in the feminist movement) is more likely to occur than one of the conjuncts (e.g., Linda is a bank teller). This belief violates the conjunction rule in probability theory. Tversky and Kahneman called this phenomenon the “conjunction fallacy”. Since the discovery of the phenomenon in 1983, researchers in psychology and philosophy have engaged in important controversies around the conjunction fallacy. The goal of this paper is to explore the most important of these controversies, namely, the controversy about the nature of the conjunction fallacy. Is the conjunction fallacy mainly due to a misunderstanding of the problem by participants (misunderstanding hypothesis) or is it mainly due to a genuine reasoning bias (reasoning bias hypothesis)? A substantial portion of research on the topic has been directed to test the misunderstanding hypothesis. I review this literature and argue that a stronger case can be made against the misunderstanding hypothesis. Thus, I indirectly provide support for the reasoning bias hypothesis.     

Quantum structure in cognition

The broader scope of our investigations is the search for the way in which concepts and their combinations carry and influence meaning and what this implies for human thought. More specifically, we examine the use of the mathematical formalism of quantum mechanics as a modeling instrument and propose a general mathematical modeling scheme for the combinations of concepts. We point out that quantum mechanical principles, such as superposition and interference, are at the origin of specific effects in cognition related to concept combinations, such as the guppy effect and the overextension and underextension of membership weights of items. We work out a concrete quantum mechanical model for a large set of experimental data of membership weights with overextension and underextension of items with respect to the conjunction and disjunction of pairs of concepts, and show that no classical model is possible for these data. We put forward an explanation by linking the presence of quantum aspects that model concept combinations to the basic process of concept formation. We investigate the implications of our quantum modeling scheme for the structure of human thought, and show the presence of a two-layer structure consisting of a classical logical layer and a quantum conceptual layer. We consider connections between our findings and phenomena such as the disjunction effect and the conjunction fallacy in decision theory, violations of the sure thing principle, and the Allais and Elsberg paradoxes in economics.     

On the reality of the conjunction fallacy

Attributing higher "probability" to a sentence of form p-and-q, relative to p, is a reasoning fallacy only if (1) the word probability carries its modern, technical meaning and (2) the sentence p is interpreted as a conjunct of the conjunction p-and-q. Legitimate doubts arise about both conditions in classic demonstrations of the conjunction fallacy. We used betting paradigms and unambiguously conjunctive statements to reduce these sources of ambiguity about conjunctive reasoning. Despite the precautions, conjunction fallacies were as frequent under betting instructions as under standard probability instructions.     

Who is susceptible to conjunction fallacies in category-based induction?

Recent evidence suggests that the conjunction fallacy observed in people’s probabilistic reasoning is also to be found in their evaluations of inductive argument strength. We presented 130 participants with materials likely to produce a conjunction fallacy either by virtue of a shared categorical or a causal relationship between the categories in the argument. We also took a measure of participants’ cognitive ability. We observed conjunction fallacies overall with both sets of materials but found an association with ability for the categorical materials only. Our results have implications for accounts of individual differences in reasoning, for the relevance theory of induction, and for the recent claim that causal knowledge is important in inductive reasoning.     

The conjunction fallacy and the many meanings of and

According to the conjunction rule, the probability of A and B cannot exceed the probability of either single event. This rule reads and in terms of the logical operator wedge, interpreting A and B as an intersection of two events. As linguists have long argued, in natural language "and" can convey a wide range of relationships between conjuncts such as temporal order ("I went to the store and bought some whisky"), causal relationships ("Smile and the world smiles with you"), and can indicate a collection of sets rather than their intersection (as in "He invited friends and colleagues to the party"). When "and" is used in word problems researching the conjunction fallacy, the conjunction rule, which assumes the logical operator wedge, therefore cannot be mechanically invoked as a norm. Across several studies, we used different methods of probing people's understanding of and-conjunctions, and found evidence that many of those respondents who violated the conjunction rule in their probability or frequency judgments inferred a meaning of and that differs from the logical operator wedge. We argue that these findings have implications for whether judgments involving ambiguous and-conjunctions that violate the conjunction rule should be considered manifestations of fallacious reasoning or of reasonable pragmatic and semantic inferences.     

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.     

Misperception of Chance,Conjunction, Framing Effects and Belief in the Paranormal: A Further Evaluation

Studies exploring relationships between belief in the paranormal and vulnerability to cognitive bias suggest that believers are liable to misperception of chance and conjunction fallacy. Research investigating misperception of chance has produced consistent findings, whilst work on conjunction fallacy is less compelling. Evidence indicates also that framing biases within a paranormal context can increase believers' susceptibility. The present study, using confirmatory factor analysis and structural equation modelling, examined the contribution of each bias to belief in the paranormal and assessed the merits of previous research. Alongside, the Revised Paranormal Belief Scale, participants completed standard and paranormal framed perception of randomness and conjunction problems. Perception of randomness was more strongly associated with belief in the paranormal than conjunction fallacy. Inherent methodological issues limited the usefulness of framing manipulations; presenting problems within a paranormal context weakened their predictive power.Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Source reliability and the conjunction fallacy

Information generally comes from less than fully reliable sources. Rationality, it seems, requires that one take source reliability into account when reasoning on the basis of such information. Recently, Bovens and Hartmann (2003) proposed an account of the conjunction fallacy based on this idea. They show that, when statements in conjunction fallacy scenarios are perceived as coming from such sources, probability theory prescribes that the "fallacy" be committed in certain situations. Here, the empirical validity of their model was assessed. The model predicts that statements added to standard conjunction problems will change the incidence of the fallacy. It also predicts that statements from reliable sources should yield an increase in fallacy rates (relative to unreliable sources). Neither the former (Experiment 1) nor the latter prediction (Experiment 3) was confirmed, although Experiment 2 showed that people can derive source reliability estimates from the likelihood of statements in a manner consistent with the tested model. In line with the experimental results, model fits and sensitivity analyses also provided very little evidence in favor of the model. This suggests that Bovens and Hartmann's present model fails to explain fully people's judgements in standard conjunction fallacy tasks.     

The degree of epistemic justification and the conjunction fallacy

This paper describes a formal measure of epistemic justification motivated by the dual goal of cognition, which is to increase true beliefs and reduce false beliefs. From this perspective the degree of epistemic justification should not be the conditional probability of the proposition given the evidence, as it is commonly thought. It should be determined instead by the combination of the conditional probability and the prior probability. This is also true of the degree of incremental confirmation, and I argue that any measure of epistemic justification is also a measure of incremental confirmation. However, the degree of epistemic justification must meet an additional condition, and all known measures of incremental confirmation fail to meet it. I describe this additional condition as well as a measure that meets it. The paper then applies the measure to the conjunction fallacy and proposes an explanation of the fallacy.     

A case of confusing probability and confirmation

Tom Stoneham put forward an argument purporting to show that coherentists are, under certain conditions, committed to the conjunction fallacy. Stoneham considers this argument a reductio ad absurdum of any coherence theory of justification. I argue that Stoneham neglects the distinction between degrees of confirmation and degrees of probability. Once the distinction is in place, it becomes clear that no conjunction fallacy has been committed.     

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.