1. Concepts of Probability: Combinatorial analysis, conditional probability, random variables and their distributions, generating functions, sum of random variables, probability modeling.
2. Stochastic Processes: Random walk, Poisson process, Markov chains,. Martingales, Borwnian motion.
3. Measure Theoretical Foundations: Probability measures, random variables, transformations of random variables, distribution functions, expectation, modes of convergence of random variables, independence, conditional expectation.
4. Law of Large Numbers: Weak and strong law of large numbers, Borel-Cantelli lemma, zero-one laws, Kolmogoroff maximal inequality, 3-series theorem, strong law of large numbers.
5. Convergence in Distribution: Weak convergence of probability measures, characteristic functions, Levy continuity theorem, central limit theorems.

The applied probability exam places special emphasis on problems and applications of the ideas and less emphasis on proofs of the theorems than the probability exam.

• Ash, R.: Real Analysis and Probability
• Billingsley, P: Probability and Measure
• Chung, K.: A Course in Probability Theory
• Dudley, R.: Real Analysis and Probability
• Durrett, R.: Probability: Theory and Examples
• Feller, W.: An Introduction to Probability Theory and Applications
• Grimmet and Welsh: Probability: An Introduction
• Karlin and Taylor: A First Course in Stochastic Processes
• Lamperti, J.: Probability
• Ross, S.: A First Course in Probability
• Ross, S.: Introduction in Probability Models