Intuitive reasoning about probability: theoretical and experimental analyses of the "problem of three prisoners"

Among various Bayesian problems of probability, the "problem of three prisoners" (Lindley, 1971; Mosteller, 1965) is an especially good example which illustrates the drastic discrepancy between intuitive reasoning and mathematical formal reasoning about probability. In particular, it raises intriguing questions concerning the mathematical and cognitive relevance of factors such as prior probabilities and the context in which certain information is given. In the current paper, we report a new version of the problem which turned out to be even more counterintuitive. This new version was also designed so that different inferential schemes would lead to separate estimates of posterior probability. The data obtained from questionnaires and theoretical analyses of the original and modified problems suggest that: (1) The psychological processes of intuitive reasoning are qualitatively different from mathematical reasoning. (2) The tendency to neglect prior probabilities (Tversky & Kahneman, 1974, 1982) is not always the critical factor for illusory judgments. (3) Intuitive judgments can be categorized by several, distinctive propositional beliefs from which the judgments are apparently derived. We call these prototypical, crude beliefs "subjective theorems," and discuss their nature and roles in the current paper.