MATH 165: Probability

MATH 165 is an introduction to probability theory, the mathematical framework for analyzing random events. The course covers core topics such as random variables, distributions, expectations, and central limit theorems, with applications ranging from games of chance to real-world modeling. It provides a strong foundation for statistics (MATH 166), data science, and actuarial science, and is relevant to fields such as science, engineering, and economics.

The course begins with the fundamentals of probability, including sample spaces, events, and basic combinatorics. We then study conditional probability and independence, followed by discrete and continuous random variables and their distributions. The concepts of expectation and variance are introduced, both for individual and jointly distributed random variables. We examine functions of random variables, conditional expectations, and key theoretical results such as Chebyshev’s inequality and the weak and strong laws of large numbers. Additional topics include moment-generating functions, the central limit theorem, and an introduction to Poisson processes.