We introduce conditional probability, independence of events, and Bayes' rule.

1. Probability and Counting
2. Story Proofs, Axioms of Probability
3. Birthday Problem, Properties of Probability
4. Conditional Probability
5. Conditioning Continued, Law of Total Probability
6. Monty Hall, Simpson's Paradox
7. Gambler's Ruin and Random Variables
8. Random Variables and Their Distributions
9. Expectation, Indicator Random Variables, Linearity
10. Expectation (Continued)
11. The Poisson distribution
12. Discrete vs. Continuous, the Uniform
13. Normal Distribution
14. Location, Scale, and LOTUS
15. Midterm Review
16. Exponential Distribution
17. Moment Generating Functions
18. MGFs (Continued)
19. Joint, Conditional, and Marginal Distributions
20. Multinomial and Cauchy
21. Covariance and Correlation
22. Transformations and Convolutions
23. Beta distribution
24. Gamma distribution and Poisson process
25. Order Statistics and Conditional Expectation
26. Conditional Expectation (Continued)
27. Conditional Expectation given an R.V.
28. Inequalities
29. Law of Large Numbers and Central Limit Theorem
30. Chi-Square, Student-t, Multivariate Normal
31. Markov Chains
32. Markov Chains (Continued)
33. Markov Chains Continued Further
34. Course Overview: A Look Ahead
35. The Soul of Statistics

This course is an introduction to probability as a language and set of tools for understanding statistics, science, risk, and randomness. The ideas and methods are useful in statistics, science, engineering, economics, finance, and everyday life. Topics include the following. Basics: sample spaces and events, conditioning, Bayes' Theorem. Random variables and their distributions: distributions, moment generating functions, expectation, variance, covariance, correlation, conditional expectation. Univariate distributions: Normal, t, Binomial, Negative Binomial, Poisson, Beta, Gamma. Multivariate distributions: joint, conditional, and marginal distributions, independence, transformations, Multinomial, Multivariate Normal. Limit theorems: law of large numbers, central limit theorem. Markov chains: transition probabilities, stationary distributions, reversibility, convergence. Prerequisite: single variable calculus, familiarity with matrices.

Statistics 110 (Probability) has been taught at Harvard University by Joe Blitzstein (Professor of the Practice in Statistics, Harvard University) each year since 2006. The on-campus Stat 110 course has grown from 80 students to over 300 students per year in that time. Lecture videos, review materials, and over 250 practice problems with detailed solutions are provided.