This text provides graduate students with a rigorous treatment of probability theory, with an emphasis on results central to theoretical statistics. It presents classical probability theory motivated with illustrative examples in biostatistics, such as outlier tests, monitoring clinical trials, and using adaptive methods to make design changes based on accumulating data. In addition, counterexamples further clarify nuances in meaning and expose common fallacies in logic. The authors explain different methods of proofs and show how they are useful for establishing classic probability results.