Probability

A book in 20 chapters about chance: from coin flips and the birthday problem to Bayes' theorem, the bell curve, Markov chains, Brownian motion and the mathematics underneath them. Each chapter starts from a question, proves what it claims, works examples step by step, and ends with questions whose answers are folded underneath.

How to read it

• In order. Each chapter needs only the ones before it; the Next link at the end of every chapter walks you through.
• Levels inside a chapter. The box at the top of each chapter says which sections are beginner, intermediate, advanced or expert. On a first read, skip the harder sections and come back later.
• Calculus from Part III. Parts I and II need only school algebra and the counting of the Combinatorics book's first chapters. From chapter 9 on, integrals appear.
• Slower help. The first chapters point to the matching lessons on the Topics shelf, which teach the same ideas more gently, for younger readers.
• Simulate. A few lines of Python can flip a million coins; many chapters show how, so you can check a result against experiment.

Contents

Part	Chapter	Level
I. Chance	1. What probability is	beginner
	2. Counting and equally likely outcomes	beginner
	3. Conditional probability and independence	beginner to intermediate
	4. Bayes' theorem	beginner to intermediate
II. Random variables	5. Discrete random variables	intermediate
	6. Expectation and variance	intermediate
	7. Inequalities and the law of large numbers	intermediate
	8. Generating functions in probability	intermediate to advanced
III. Continuous probability	9. Continuous random variables	intermediate
	10. The normal distribution	intermediate
	11. Joint distributions	intermediate to advanced
	12. The central limit theorem	advanced
IV. Random processes	13. Markov chains	advanced
	14. Poisson processes	advanced
	15. Random walks and martingales	advanced
	16. Brownian motion	advanced to expert
V. Foundations and frontier	17. Measure-theoretic probability	expert
	18. Information and entropy	advanced to expert
	19. Randomness in computing	advanced
	20. Where probability goes	expert

The glossary defines every term, with a link to the chapter that teaches it.

Three ways in

• Starting out: read chapters 1 to 7. They are the probability most people ever need, with the reasons written out.
• You know calculus: read chapters 9 to 12 for the normal curve and the central limit theorem, then 13 to 16 for random processes.
• You want the foundations: chapter 17, then 18 to 20.