An exposition of possibility and probability

TL;DR

This paper explores the conceptual relationship between possibility and probability, especially zero-probability events, using modal logic and mathematical frameworks, clarifying when they align or differ.

Contribution

It introduces a formal framework linking possibility and probability, analyzing zero-probability events and their implications in probabilistic models.

Findings

Conditions where possibility and probability diverge

Conditions where possibility and probability are equivalent

Application to multinomial models

Abstract

This paper considers the notion of possible events which are insignificant in probabilistic analysis (i.e. events that have zero probability). The paper discusses the method of modal logic based on "possible worlds" and discusses a mathematical framework for the concepts of possibility, impossibility and certainty that are sometimes (incorrectly) thought to be defined with respect to probability. The relationship between possibility and probability is explored for general probability spaces and for refinements of these spaces conditional on other events, with particular focus on the properties of events having zero probability. We derive conditions under which possibility and significance diverge and conditions under which they can be reconciled as equivalent ideas within certain contexts. We also apply this analysis to discuss issues in possibility and probability in the multinomial model.