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

Abstract

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.

The Bayesian sampler: Generic Bayesian inference causes incoherence in human probability judgments.

“…The rate of random error is enhanced from d (for single events) to d + ∆d (for conjunctions and disjunctions). This assumption is justified on the basis that combined variables (i.e., conjunctions and disjunctions) will be noisier than individual variables (Costello & Watts, 2017). This is also a necessary assumption for the PT+N model to predict above-chance rates of conjunction fallacy -if the noise is higher for conjunctions, then the mean estimates for a conjunction could be higher than the mean estimates of the simple events because conjunctions are more strongly regressed towards 0.5 (Costello & Watts, 2017).…”

An account of subjective probability judgment for joint events: Conjunctive and disjunctive

“…Conversely, we found that the other component probability accounted for more of the variance than was the case in the normative model where this component accounted for only 1.4–2%. Since the Costello and Watts () model predicts that estimated probabilities are linear transformations of the actual objective probabilities, a simulation revealed that their model yields similar percentages to the normative model: 54.3–54.8% for the reference point probability and 1.6–1.9% for the other component. The proportions predicted by the Nilsson et al .…”

“…Interestingly, it can be shown that under standard conditions in both Costello and Watts’ () model (in which judgments are essentially normative but subject to systematic but mutually self‐canceling errors) and Nilsson and co‐workers’ (2009) model (where joint probabilities are said to be weighted averages of the component probabilities), the two joint event‐component probability differences (conjunctive‐less likely, more likely – disjunctive) should be correlated. From the results that we present in Studies 1 and 2 we see that this was not the case.…”

“…Relative to the conjunction fallacy, violations of the disjunction rule have been relatively neglected in the research literature (Cf. Bar‐Hillel & Neter, ; Carlson & Yates, ; Costello & Watts, ; Nilsson, Winman, Juslin & Hansson, ). The present paper seeks to evaluate the processes underpinning conjunctive and disjunctive judgments within a unified framework and in the process, we develop our own model of joint probability judgment.…”

“…Indeed, Gavanski and Roskos‐Ewoldsen () demonstrated that while single event probabilities might be conceivably based on degrees of representativeness, the conjunctive event probability did not seem to be so. While more recent theoretical perspectives including those of Costello and Watts () and Nilsson et al . () share our focus on the relationship between the component events and the corresponding joint events, in our case, we have directly examined this relationship using statistical means including regression analysis, within participants ANOVA and more recently multi‐level linear modeling.…”