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This lecture covers the fundamental concepts of applied probability and stochastic processes, focusing on topics such as transition probability matrices, Markov chains, and characteristic polynomials. The instructor explains the n-step transition probability matrix, eigenvalues, and linear combinations of matrices. The lecture also delves into communication classes, recurrent and transient states, and the Cayley-Hamilton theorem. Various exercises explore random walks, invariant distributions, and expected return times in different scenarios.