# Foundations for conditional probability

## TL;DR

This paper presents a foundational perspective on probability, characterizing it as a subset of conditional expectation derived from a plausible preorder on random variables, extending traditional definitions to undefined or zero-probability cases.

## Contribution

It introduces a novel characterization of probability through plausible preorders, broadening the scope of conditional probability beyond classical rules.

## Key findings

- Probability is characterized as a subset of conditional expectation.
- The approach extends conditional probability to cases with zero or undefined probabilities.
- Provides a new perspective on the foundations of probability theory.

## Abstract

The main result presented in this article is that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities. This is justified by the fact that probability is coherent as confirmed by its common formalizations, and by our result that a function is coherent if and only if it is a subset of conditional expectation induced by a plausible preorder on random quantities. In addition to offering a different perspective on conditional probability, our use of a plausible preorder in the role of a fundamental notion extends conditional probability to cases in which the calculation of conditional probability using the P(A|C)=\frac{P(A\wedge C)}{P(C)} rule fails: if P is a coherent function, then it can be extended so that for every event A and nonzero event C holds that P(A|C)=0 if A\wedge C=0 and P(A|C)=1 if A\wedge C=C, no matter whether the unconditional probability P(C) is zero or whether it is defined.