It is easy to construct pairs of sentences X, Y that lead many people to ascribe higher probability to the conjunction X-and-Y than to the conjuncts X, Y. Whether an error is thereby committed depends on reasoners’ interpretation of the expressions “probability” and “and.” We report two experiments designed to clarify the normative status of typical responses to conjunction problems. © 2004 Cognitive Science Society, Inc. All rights reserved.

We have been oversold on the base rate fallacy in probabilistic judgment from an empirical, normative, and methodological standpoint. At the empirical level, a thorough examination of the base rate literature (including the famous lawyer–engineer problem) does not support the conventional wisdom that people routinely ignore base rates. Quite the contrary, the literature shows that base rates are almost always used and that their degree of use depends on task structure and representation. Specifically, base rates play a relatively larger role (...) in tasks where base rates are implicitly learned or can be represented in frequentist terms. Base rates are also used more when they are reliable and relatively more diagnostic than available individuating information. At the normative level, the base rate fallacy should be rejected because few tasks map unambiguously into the narrow framework that is held up as the standard of good decision making. Mechanical applications of Bayes's theorem to identify performance errors are inappropriate when (1) key assumptions of the model are either unchecked or grossly violated, and (2) no attempt is made to identify the decision maker's goals, values, and task assumptions. Methodologically, the current approach is criticized for its failure to consider how the ambiguous, unreliable, and unstable base rates of the real world are and should be used. Where decision makers' assumptions and goals vary, and where performance criteria are complex, the traditional Bayesian standard is insufficient. Even where predictive accuracy is the goal in commonly defined problems, there may be situations (e.g., informationally redundant environments) in which base rates can be ignored with impunity. A more ecologically valid research program is called for. This program should emphasize the development of prescriptive theory in rich, realistic decision environments. (shrink)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

This paper uses the framework of Popper and Miller's work on axiom systems for conditional probabilities to explore Adams' thesis concerning the probabilities of conditionals. It is shown that even very weak axiom systems have only a very restricted set of models satisfying a natural generalisation of Adams' thesis, thereby casting severe doubt on the possibility of developing a non-Boolean semantics for conditionals consistent with it.

Subjects were requested to choose between gambles, where the outcome of one gamble depended on a single elementary event, and the other depended on an event compounded of a series of such elementary events. The data supported the hypothesis that the subjective probability of a compound event is systematically biased in the direction of the probability of its components resulting in overestimation of conjunctive events and underestimation of disjunctive events. Studies pertaining to this topic are discussed.

This book offers a concise survey of basic probability theory from a thoroughly subjective point of view whereby probability is a mode of judgment. Written by one of the greatest figures in the field of probability theory, the book is both a summation and synthesis of a lifetime of wrestling with these problems and issues. After an introduction to basic probability theory, there are chapters on scientific hypothesis-testing, on changing your mind in response to generally uncertain observations, on expectations of (...) the values of random variables, on de Finetti's dissolution of the so-called problem of induction, and on decision theory. (shrink)

It is easy to construct pairs of sentences X, Y that lead many people to ascribe higher probability to the conjunction X‐and‐Y than to the conjuncts X, Y. Whether an error is thereby committed depends on reasoners' interpretation of the expressions “probability” and “and.” We report two experiments designed to clarify the normative status of typical responses to conjunction problems.

We propose a theory that relates perceived evidence to numerical probability judgment. The most successful prior account of this relation is Support Theory, advanced in Tversky and Koehler. Support Theory, however, implies additive probability estimates for binary partitions. In contrast, superadditivity has been documented in Macchi, Osherson, and Krantz, and both sub- and superadditivity appear in the experiments reported here. Nonadditivity suggests asymmetry in the processing of focal and nonfocal hypotheses, even within binary partitions. We extend Support Theory by revising (...) its basic equation to allow such asymmetry, and compare the two equations’ ability to predict numerical assessments of probability from scaled estimates of evidence for and against a given proposition. Both between- and within-subject experimental designs are employed for this purpose. We find that the revised equation is more accurate than the original Support Theory equation. The implications of asymmetric processing on qualitative assessments of chance are also briefly discussed. (shrink)

We advance a theory of inductive reasoning based on similarity, and test it on arguments involving mammal categories. To measure similarity, we quantified the overlap of neural activation in left Brodmann area 19 and the left ventral temporal cortex in response to pictures of different categories; the choice of of these regions is motivated by previous literature. The theory was tested against probability judgments for 40 arguments generated from 9 mammal categories and a common predicate. The results are interpreted in (...) the context of Hume’s thesis relating similarity to inductive inference. (shrink)