This is a course on the fundamentals of probability geared towards first- or second-year graduate students who are interested in a rigorous development of the subject. The course covers most of the topics in 6.431 (sample space, random variables, expectations, transforms, Bernoulli and Poisson processes, finite Markov chains, limit theorems) but at a faster pace and in more depth. There are also a number of additional topics, such as language, terminology, and key results from measure theory; interchange of limits and expectations; multivariate Gaussian distributions; deeper understanding of conditional distributions and expectations.