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This book is intended for students who have completed proof-based courses in Advanced Calculus and Linear Algebra. In addition to the standard topics (such as the Poisson approximation to the binomial, law of large numbers, central limit theorem, Markov chains, and simple linear regression), several other topics and results that are accessible at this level and that fit into a one semester course are covered: -- first moment method with some applications, such as cliques in the Erdos-Renyi random graph, and an upper bound on the typical longest increasing subsequence of a random permutation; -- second moment method with applications to Bernstein's polynomials, cliques in the Erdos-Renyi random graph, and the Hardy-Ramanujan theorem; -- Hoeffding's inequality, and the Johnson-Lindenstrauss lemma; -- the Hoeffding-Chernoff inequality, and the generalization ability of classification algorithms; -- Azuma's inequality with several examples, such as the chromatic number of the Erdos-Renyi random graph, max-cut in sparse random graphs, and the Hamming distance on the hypercube. A knowledge of Lebesgue integration is not assumed, although the discussion of continuous distributions gives some idea of why learning about it is something to look forward to. A number of exercises are included throughout and at the end of each section. In the second edition, a number of explanations were clarified and some mistakes were corrected.