Topics: 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

1. Probability, Intuition, and Axioms
2. Probability by Counting and Inclusion-Exclusion
3. Principles of Counting
4. Conditional Probability
5. Conditional Craps
6. Lying Witnesses and Simpson's Paradox
7. Random Variables & Distributions
8. Expectation I: Binomial Expectation & Variance
9. Expectation II: Infinite & Conditional Expectation
10. Geometric & Negative Hypergeometric Distributions
11. Gambling: Random Walks & Gambler's Ruin
12. Expected Lead Time & Bijections between Paths
13. Variables: Independent, Uncorrelated, & Generating Functions
14. Basic Inequality, Markov's Inequality & Chebyshov's Inequality
15. Review and Questions #1-10

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.