Ankur Moitra, teaching MIT's 18.200 Principles of Discrete Applied Mathematics, proves Chernoff's theorem for bounding the tails of sums of independent random variables. The lecture builds on earlier tail-bound results covered in the course, such as Markov and Chebyshev inequalities, and shows how independence lets you get exponentially tighter bounds on how far a sum of random variables can stray from its expectation. Moitra works through the derivation at the board, setting up the moment generating function approach and optimizing over a free parameter to get the sharpest bound. The lecture is dense with algebraic manipulation but keeps returning to the intuition behind each step, explaining why independence is the ingredient that makes such strong concentration possible. Part of a full-semester course on discrete probability and combinatorics methods used throughout computer science and applied mathematics.