Chapter 1: The Axioms of Probability

Learning Objectives

• Construct a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ from a description of an experiment
• Identify the appropriate sigma-algebra for finite, countably infinite, and uncountable sample spaces
• State and apply Kolmogorov's three axioms to derive all elementary consequences
• Prove the inclusion-exclusion formula and the union bound and apply them to communications problems
• State the continuity of probability and both Borel-Cantelli lemmas
• Compute combinatorial probabilities using the four sampling paradigms
• Apply the classical probability model to finite uniform spaces in communications settings