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Sets, Counting, and Probability

Online math lectures with Harvard professor

Paul G. Bamberg, DPhil, Senior Lecturer on Mathematics, Harvard University

This online math course develops the mathematics needed to formulate and analyze probability models for idealized situations drawn from everyday life. Topics include elementary set theory, techniques for systematic counting, axioms for probability, conditional probability, discrete random variables, infinite geometric series, and random walks. Applications to card games like bridge and poker, to gambling, to sports, to election results, and to inference in fields like history and genealogy, national security, and theology. The emphasis is on careful application of basic principles rather than on memorizing and using formulas.

Free online math lectures

The Quicktime and MP3 formats are available for download, or you can play the Flash version directly. Each lecture is 2 hours.

Probability, Intuition, and Axioms

  • Video/Audio
    • Probability by Symmetry; Probability by Experiment; Payoffs and Probability; Fair Price and Probability; Notation to Combine Sets; Venn Diagrams; Events and Sample Spaces; Event Spaces; Addition of Probabilities; Probability Functions; Inclusion-Exclusion Rule; Many Ways to Skin a Cat
  • Class Materials

Probability by Counting and Inclusion-Exclusion

  • Video/Audio
    • Review of the Basics; DeMorgan's Laws; Inclusion-Exclusion; Cups and Saucers; Cups and Saucers # 2; Urns; Family Planning; Twenty-first-Century Sheherezade; Craps; Achilles Rolls of a 6
  • Class Materials

Principles of Counting

  • Video/Audio
    • Count A Cartesian Product; The Binomial Theorem; Counting Poker Hands; Counting Bridge Hands; The Munchkin Problem; Genoese Lottery; Counting Suit Patterns; Negative Binomial Theorem; Count by Inclusion-Exclusion
  • Class Materials

Conditional Probability

  • Video/Audio
    • St. Paul at Lystra; Conditional Four Aces; Conditional Probability; Independence of Events; Complimentary Events; The Bearded Man Problem; Lemons; Two Useful Theorems; Conditional Munchkins; Monty Hall Problem; Random Desserts
  • Class Materials

Conditional Craps

Lying Witnesses and Simpson's Paradox

Random Variables

  • Video/Audio
    • Discrete Random Variables; Event Space for a DRV; Bridge and Poker; Binomial Distribution; Geometric Distribution; Negative Binomial Distribution; Exponential Function; Poisson Distribution; Distribution Function; Distribution Examples; Function of Random Var; First Computer Project
  • Class Materials

Expectation I

  • Video/Audio
    • Unconscious Statistician; Properties of Expectation; Binomial Expectation; Tail-Sum Theorem; Geometric Tail-Sum; Negative Binomial Expectation; Coupons Problem; Variance; Telescoping Series; Convergence
  • Class Materials

Expectation II

Tartan Dice; Terminated Geometric; World Series Pitchers; Negative Hypergeometric; Coin Tossing

Gambling

  • Video/Audio
    • Casino Night; Random Walk Examples; Random Walk Terminology; Gambler's Ruin; Ruin in Fair Casino; Time until Win or Ruin; A Fair Game; Walk as a Random Variable; Ballot Theorem; Hitting Time Theorem; Expected Lead Times
  • Class Materials

Expected Lead Time; Bijections between Paths

  • Video/Audio

Variables

  • Video/Audio
    • First Visit to Max; Sojourn Times; Independent Variables; Uncorrelated Variables: A Counterexample; Generating Function; Product of Gen Functions; A Simple Example; Gen Function for 2 Dice; Clever Loaded Dice; Well-Known Distributions
  • Class Materials

Inequality

Questions #1–10

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