After taking the course, the student should be able to show that she or he has experienced the width of probability theory and its applications. The course will include but not limited to construction of probability spaces and random variables, integration, conditional expectation, law of large numbers, convergence of series, weak convergence,  characteristic functions and central limit theorems, random walk, convergence of random variables, modes of convergence, introduction of martingales, martingale convergence theorem higher order moments: kurtosis and skewness.