Intuitive probability of non-intuitive events

Abstract

Quantitative probability in the subjective theory is assumed to be finitely additive and defined on all the subsets of an underlying state space. Functions from this space into an Euclidian n-space create a new probability space for each such function. We point out that the associated probability measures, induced by the subjective probability, on these new spaces can not be finitely additive and defined on all the subsets of Euclidian n-space, for n ≥ 3. This is a consequence of the Banach-Tarski paradox. In the paper we show that subjective probability theory, including Savage’s theory of choice, can be reformulated to take this, and similar objections into account. We suggest such a reformulation which, among other things, amounts to adding an axiom to Savage’s seven postulates, and then use a version of Carathéodory’s extension theorem.