De Finetti for mathematics undergraduates

TL;DR

This paper presents a simplified, accessible proof of de Finetti's foundational theorems in probability theory, including the Dutch Book Theorem and exchangeability, suitable for undergraduates with basic mathematical background.

Contribution

It provides elementary proofs of de Finetti's key theorems within boolean algebras, making advanced probability concepts accessible to undergraduates.

Findings

Proved de Finetti's theorem using boolean algebra

Established the Fundamental Theorem of Probability in an elementary way

Provided a combinatorial proof of exchangeability theorem

Abstract

In 1931 de Finetti proved what is known as his Dutch Book Theorem. This result implies that the finite additivity {\it axiom} for the probability of the disjunction of two incompatible events becomes a {\it consequence} of de Finetti's logic-operational consistency notion. Working in the context of boolean algebras, we prove de Finetti's theorem. The mathematical background required is little more than that which is taught in high school. As a preliminary step we prove what de Finetti called ``the Fundamental Theorem of Probability'', his main contribution both to Boole's probabilistic inference problem on the object of probability theory, and to its modern reformulation known as the optimization version of the probabilistic satisfiability problem. In a final section, we give a self-contained combinatorial proof of de Finetti's exchangeability theorem.