Foundations of Discrete Probability and the Probabilistic Method

Probability theory provides the mathematical backbone for randomized algorithms, data structures, and modern theoretical computer science. This text-based course guides you through core probability principles and the probabilistic method, teaching you how to analyze discrete systems and construct rigorous proofs. You will build confidence working with random variables, expectation, and foundational inequalities used throughout discrete mathematics and algorithm design. What you'll learn: - Understand fundamental probability spaces, events, and essential axioms - Apply linearity of expectation to simplify complex combinatorial calculations - Utilize Markov's inequality and probabilistic bounds to analyze algorithmic behavior - Master the probabilistic method to prove the existence of combinatorial objects - Explore classical applications in Boolean logic, circuit complexity, and satisfiability - Practice constructing formal mathematical proofs through structured written exercises The material begins with key terminology and basic definitions before moving into expectation, probabilistic bounding techniques, and practical combinatorial applications. This course is created for beginners in discrete mathematics and computer science theory. No advanced background in probability is required to get started. Begin reading to develop your mathematical toolkit and master the probabilistic method.