The course
Learn by technique
Eight parts, from counting to brainteasers. Every section opens with a short toolkit, then its problems from easiest to hardest.
Part I
Counting
Counting arguments that every later chapter relies on.
59 problems · 2 chapters
1 Counting Fundamentals
The product rule, permutations and combinations, and their two workhorse extensions.
2 Clever Counting
When direct enumeration is hopeless: bijections, paths, recursions and symmetry.
Part II
Probability
Probabilities of events, from equally likely outcomes to continuous and geometric models.
193 problems · 7 chapters
3 Equally Likely Outcomes
Probability as favourable over total, computed by counting.
4 Symmetry
Finding an equally likely structure hidden in the problem, so that no calculation is needed.
5 Conditional Probability and Bayes
Updating probabilities on information, and the traps that come with it.
6 First-Step Analysis and Races
Condition on what happens first and solve the resulting equations.
7 Discrete Distributions
The standard discrete families and the arguments that identify them.
8 Continuous Distributions
Densities, transformations and the uniform, normal and exponential families.
9 Geometric Probability
When the probability is an area, a length or an angle.
Part III
Expectation and Variance
Expected values, conditional expectations, extremes and second moments.
125 problems · 5 chapters
10 Expectation Basics and Linearity
Expected values from the definition, then the single most useful trick in the book.
11 Waiting Times
How long until something happens.
12 Conditional Expectation
Expectation given partial information, and the tower property.
13 Extremes and Order Statistics
Expected maxima, minima, spacings and records.
14 Variance, Covariance and Moments
Second moments, correlation and generating functions.
Part IV
Random Processes
Markov chains, random walks, martingales, Poisson processes, Brownian motion and Ito calculus.
44 problems · 4 chapters
15 Markov Chains
Memoryless state machines: hitting times, long-run behaviour, branching.
16 Random Walks and Martingales
Simple walks on the integers and beyond, and the fair-game martingales that tame them.
18 Brownian Motion and Ito Calculus
The continuous-time Gaussian random walk, Ito's lemma and its consequences.
Part V
Statistics and Regression
Estimation, testing and least squares.
15 problems · 2 chapters
19 Estimation and Testing
Estimating parameters and deciding between hypotheses.
20 Linear Regression
Ordinary least squares and what happens when the data change.
Part VI
Games and Decisions
Choosing well under uncertainty, alone or against an opponent.
48 problems · 3 chapters
21 Expected-Value Decisions
One or a few decisions, each made by comparing expected values.
22 Optimal Stopping
When to stop a sequential game.
23 Strategic Games
Games against a rational opponent.
Part VII
Finance and Derivatives
Arbitrage, options, pricing and hedging.
31 problems · 3 chapters
24 Arbitrage and Portfolios
Riskless profit and the arithmetic of combining assets.
25 Option Payoffs, Parity and Replication
Model-free facts about options.
26 Option Pricing and Greeks
Models that put a number on an option, and the sensitivities traders hedge.
Part VIII
Brainteasers and Pure Mathematics
Logic, search, construction, number theory, algebra, calculus and geometry.
158 problems · 8 chapters
27 Logic and Deduction
Reasoning puzzles with no computation.
28 Information, Search and Weighing
How much each test can tell you.
29 Constructions and Scheduling
Show that it can be done, and do it in the fewest steps.
30 Invariants, Extremal Arguments and Recursion
Structural arguments that replace computation.
31 Number Theory and Digits
Divisibility, congruences and digit puzzles.
32 Algebra and Word Problems
Setting up and solving the right equations.
33 Calculus, Series and Linear Algebra
Analysis and matrices as they appear in interviews.