An exposition of possibility and probability

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.