"After learning this course, students should be able1.to deeply understand and master the basic concepts and conclusions of modern probability theory, not only to remember these basic concepts and the basic probability laws including conditions and conclusions, but also deeply to understand?the basic principles and ideas of modern probability;2．to fully master the four basic convergence theorems (Monotone Convergence Theorem, Fatou Lemma, Dominated Convergence Theorem, and Bounded Convergence Theorems) and be able to apply them in many important topics and different problems; 3．to clearly understand the probability meaning, difference, and relationships of several kind of convergence concepts (almost everywhere convergence; convergence in measure/probability; Convergence in Lp Norm; Weak Convergence) and be able to apply them in different problems; 4．to fully master the very important concepts of conditional expectations and conditional probabilities and to improve the ability of solving practical problems by applying the basic probability methods of “conditioning”. 5. to clearly understand and master the basic concepts regarding martingales including the existence, uniqueness, properties and applications of martingales, super- and sub-martingales and be able to apply the important martingale method in the study of modern theory of stochastic processes, stochastic analysis and financial mathematics."