This course offers a rigorous introduction to probability theory with applications to computational mathematics. Emphasis is placed on nonasymptotic properties of probability models, rather than classical limit theorems. Topics include measure theory, integration, product measures, probability spaces, random variables and expectation, moments, Lp spaces, orthogonality, independence, concentration inequalities, distances between probability measures, the Berry-Esseen theorem, conditional expectation, and conditioning for Gaussian families.