Homeworks: there will be five homework sets, as well as a take-home final exam in each semester.
After a summary of the necessary results from measure theory, we will learn the probabilist's lexicon (random variables, independence, etc.). We will then develop the necessary techniques (Borel Cantelli lemmas, estimates on sums of independent random variables and truncation techniques) to prove the classical laws of large numbers. Next comes Fourier techniques and the Central Limit Theorem, followed by combinatorial techniques and the study of random walks.
"Probability: theory and examples", 3rd Edition, by R. Durrett.