Jordan Ellenberg's How Not to Be Wrong opened doors. These essays walk through them — exploring probability, game theory, statistical traps, chaos, networks, information theory, fairness, and the cognitive biases that make us human. Every essay includes working simulations, calculators, and games.
Part I: The Bets You Make
Mathematics of risk, ruin, and the irrational mind.
The Kelly Criterion
Why knowing how much to bet matters more than knowing what to bet on. The formula that keeps you from going broke even when you're right.
Ergodicity
The difference between the average and the actual. Why GDP per capita lies about your life, and what Ole Peters proved to the economists.
Gambler's Ruin
The mathematical certainty that the house wins. Not luck — the asymmetry of absorbing barriers.
Prospect Theory
Why cab drivers quit early on good days and work late on bad ones. Kahneman and Tversky's demolition of homo economicus.
Risk vs Uncertainty
Frank Knight's forgotten distinction: risk you can calculate, uncertainty you can't. The Ellsberg paradox, the 2008 crisis, and when to put down the calculator.
Part II: When to Decide
The mathematics of optimal stopping, belief updating, and impossible voting systems.
Bayes' Theorem
The theorem that reads your mind and catches your cancer. Why a 99% accurate test doesn't mean 99% probability.
Optimal Stopping
When to stop looking and start choosing. The 37% rule for hiring, dating, and apartment hunting.
Arrow's Impossibility Theorem
Democracy is mathematically doomed — but vote anyway. Why no voting system satisfies three reasonable conditions.
Part III: Hidden Structure
Patterns that emerge from numbers, from Benford's Law to the birthday problem.
Pi & Randomness
Are the digits of π random? The deepest question about the most famous number. Normal numbers, Champernowne's constant, and what "random" means for something deterministic.
Benford's Law
The law that catches liars. How the first digit of a number reveals fraud, from Enron to election audits.
Power Laws
When the average doesn't exist. Why one earthquake can release more energy than all others combined.
The Birthday Problem
The coincidence that isn't. Why 23 people have a 50% chance of a shared birthday — and why this breaks encryption.
Regression to the Mean
The most important statistical concept you've never heard of. Why the Sports Illustrated cover jinx isn't a jinx.
Part IV: The Data Lies
When aggregation reverses trends, selection creates phantom correlations, and the dead don't talk.
Simpson's Paradox
When the data tells two opposite stories. Berkeley's 1973 admissions scandal and when to aggregate vs. disaggregate.
Survivorship Bias
The dead don't talk. Abraham Wald's armor problem, mutual fund illusions, and why you don't hear about failed dropouts.
Berkson's Paradox
The phantom correlation. Why attractive people seem to have terrible personalities and restaurants are food OR ambiance.
The Inspection Paradox
Why the bus is always late, your friends have more friends than you, and class sizes are larger than advertised.
Part V: Systems That Eat Themselves
When measures become targets, and the old outlast the new.
Part VI: The Shape of Chance
Probability and randomness — the patterns hiding inside coin flips, doors, and infinite games.
The Monty Hall Problem
Why switching doors doubles your odds, and why your brain refuses to believe it. The gameshow puzzle that fooled a thousand PhDs.
The Gambler's Fallacy
The roulette wheel has no memory, but you do. Why "due" isn't a thing and streaks mean less than you think.
The St. Petersburg Paradox
A game worth infinite money that no one would pay $25 to play. The 300-year-old puzzle that birthed expected utility theory.
Streaks & the Hot Hand
When is a pattern real and when is it just noise? The hot hand debate that reversed itself after 30 years.
Martingale Theory
The mathematically perfect betting system that guarantees ruin. Why doubling down always sounds good and always ends badly.
The Law of Large Numbers
Why casinos always win and insurance companies rarely lose. The theorem that turns randomness into certainty — eventually.
The Normal Distribution
The bell curve that rules the world. From Gauss's astronomy errors to IQ tests to stock returns — why it shows up everywhere, and when it catastrophically doesn't.
The Poisson Distribution
The law of rare events. Prussian soldiers kicked by horses, V-2 bombs over London, and the beautiful math that counts the unlikely.
Parrondo's Paradox
Two losing games that become a winning strategy when combined. The counterintuitive math connecting ratchets, evolution, and portfolio theory.
Part VII: The Geometry of Choice
Game theory and decision theory — when rational agents collide, cooperate, and confound each other.
The Prisoner's Dilemma
Why rational self-interest makes everyone worse off. The Cold War game that explains climate change, doping, and price wars.
Nash Equilibrium
The beautiful mind's ugliest insight. When nobody can do better by changing strategy — even if everyone's miserable.
Tragedy of the Commons
When everyone's rational decision destroys the shared resource. Hardin's pasture, overfished oceans, and carbon emissions.
Mechanism Design
How to build games people can't cheat at. The reverse game theory behind auctions, kidney exchanges, and school choice.
Newcomb's Problem
The philosophy puzzle that splits rational people in half. One box or two? Your answer reveals your decision theory.
Braess's Paradox
When adding a road makes everyone's commute longer. The counterintuitive math of selfish routing and network design.
Part VIII: Numbers That Lie
Statistical traps and fallacies — six ways data misleads even the careful.
Base Rate Neglect
Why rare diseases produce mostly false positives. The probability error that trips up doctors, judges, and jurors.
The Ecological Fallacy
When what's true of groups is false of individuals. How aggregate data creates correlations that don't exist at the individual level.
The Texas Sharpshooter Fallacy
Painting the bullseye after you shoot. P-hacking, the Bible Code, and why testing enough hypotheses guarantees a "discovery."
The McNamara Fallacy
When you measure what's easy instead of what matters. Body counts in Vietnam and engagement metrics in Silicon Valley.
The Will Rogers Phenomenon
How moving patients between groups "cures" cancer — without helping anyone. Stage migration and the statistical illusion of progress.
Stein's Paradox
Why a baseball player's batting average predicts Tokyo's rainfall. The most counterintuitive result in statistics.
Part IX: Growth, Decay & Time
Dynamics and long-run behavior — compound growth, S-curves, chaos, and tail risk.
Compound Interest
Einstein's (alleged) eighth wonder and its dark twin: compound debt. The Rule of 72, Benjamin Franklin's experiment, and the birth of e.
Logistic Growth
Why everything that grows exponentially eventually stops. The S-curve from lily pads to COVID to startup valuations.
Chaos Theory
The butterfly, the weather, and the limits of prediction. How Edward Lorenz discovered that determinism doesn't mean predictability.
Heavy Tails & Black Swans
Why Gaussian thinking fails in Extremistan. Taleb's turkey, 25-sigma events, and the distributions that don't have averages.
Benford's Law II: The Forensic Frontier
How digit patterns catch tax cheats and election fraud. Second-digit analysis, MAD tests, and the limits of forensic mathematics.
The Ergodic Hypothesis in Economics
Why GDP growth doesn't mean you're getting richer. Ole Peters's argument that economics has been making an ergodicity error for 300 years.
Part X: Networks & Complexity
Systems thinking — small worlds, scale-free networks, segregation, and the limits of parallelism.
Six Degrees of Separation
Small world networks and why Kevin Bacon connects to everyone. Milgram's letters, weak ties, and Watts-Strogatz rewiring.
Metcalfe's Law
Why networks are worth the square of their users — and why that's dangerously wrong. N², N·log(N), and the tipping point.
Preferential Attachment
How the rich get richer in networks, science, and language. The Matthew Effect, Barabási-Albert, and why hubs dominate.
Voting Paradoxes Beyond Arrow
Gerrymandering, strategic voting, and the deeper math of manipulation. The sequel to Arrow's Impossibility Theorem.
Schelling Segregation
How mild preferences create extreme outcomes. The cellular automaton that explains residential segregation without assuming bigotry.
Amdahl's Law
Why throwing more people at a project makes it slower. Serial bottlenecks, Brooks's Law, and the limits of parallelism.
Network Effects
Why the fax machine was useless until everyone had one. Direct vs indirect network effects, critical mass, and winner-take-all dynamics.
Part XI: The Human Equation
Behavioral mathematics — how our brains systematically betray us, and what the numbers say about it.
The Dunning-Kruger Effect
The mathematics of not knowing what you don't know. McArthur Wheeler's lemon juice, the viral myth, and the statistical artifact underneath.
Anchoring & Adjustment
Why the first number you see hijacks your brain. Rigged wheels, salary negotiations, and the dice rolls that change prison sentences.
The Paradox of Choice
When more options make you worse off. Iyengar's jam study, maximizers vs. satisficers, and the math of decision paralysis.
Part XII: The Infinite and the Impossible
Pure math paradoxes — when infinity breaks intuition and logic eats itself.
Zeno's Paradoxes
Why motion is mathematically impossible — and why calculus saves us. Achilles, the tortoise, and 2,400 years of philosophical confusion resolved by limits.
Cantor's Different Infinities
Some infinities are bigger than others. The diagonal argument that proved the reals outnumber the rationals — even though both are infinite.
Gödel's Incompleteness Theorems
Why math can't prove everything about itself. The 1931 bombshell that shattered Hilbert's dream and revealed the limits of formal systems.
The Banach-Tarski Paradox
How to turn one sphere into two using nothing but the Axiom of Choice. The theorem that makes physicists nervous and set theorists smile.
Russell's Paradox
The set of all sets that don't contain themselves. The crisis that nearly destroyed mathematics and the type theory that saved it.
The Halting Problem
Why no computer can predict all computers. Turing's 1936 proof that some problems are provably unsolvable — by anything, ever.
Part XIII: Algorithms for Life
Computational thinking — the math behind optimization, search, randomness, and the algorithms that run the world.
The Traveling Salesman Problem
Finding the shortest route through N cities. Why "good enough" saves UPS $400 million a year when "perfect" would take longer than the universe.
P vs NP
The million-dollar question: is checking easier than solving? If P=NP, cryptography breaks and creativity becomes mechanical.
Explore vs Exploit
The multi-armed bandit and when to try something new. Should you go to your favorite restaurant or try the new place? Math has an answer.
Sorting Algorithms
Why your bookshelf teaches algorithm design. Bubble sort is terrible, quicksort is practical, and the information-theoretic bound says you can't do better than O(n log n).
Monte Carlo Methods
How randomness solves problems logic can't. From Buffon's needle to nuclear chain reactions — throw enough darts and π appears.
Random Walks
The drunkard's path and the stock market. A drunk man will find his way home, but a drunk bird may be lost forever.
Part XIV: Measuring the Unmeasurable
Statistics deep cuts — the theorems and techniques that power modern data science.
The Central Limit Theorem
Why everything is bell-shaped. Average enough of anything and you get a Gaussian — the "supreme law of Unreason."
Correlation vs Causation
The most abused phrase in science. Nicolas Cage films correlate with drownings, but Bradford Hill's criteria cut through the noise.
Bayesian vs Frequentist
The statistics war that won't end. Same data, different frameworks, potentially opposite conclusions. The Jeffreys-Lindley paradox.
The Bootstrap
Pulling yourself up by statistical straps. Efron's 1979 revolution: resample your data 10,000 times and uncertainty quantification is free.
Maximum Likelihood
Finding the most probable explanation. Fisher's framework powers everything from regression to neural networks to DNA sequencing.
Confidence Intervals
What they are, what they aren't, and why 97% of researchers get them wrong. The most misunderstood concept in statistics.
Overfitting
When your model memorizes the noise. Why a perfect fit to training data is the worst kind of prediction — and how regularization, cross-validation, and Occam save us.
Signal vs Noise
Finding the music in the static. Nate Silver's framework, Wiener filters, and why most of what you think is signal is actually noise.
Part XV: Markets, Madness & Money
Economics and incentives — when information asymmetry, moral hazard, and irrationality shape markets.
Adverse Selection
The market for lemons and why used car markets collapse. Akerlof's paper was rejected as "trivial" before winning the Nobel Prize.
Moral Hazard
Why bailouts make banks riskier. If you're insured against consequences, you take more risks — and everyone else pays.
The Winner's Curse
Why the winning bidder almost always overpays. Winning an auction is evidence you were wrong about the value.
Comparative Advantage
Why LeBron James shouldn't mow his own lawn. Ricardo's 1817 insight that Samuelson called "the most counterintuitive result in economics."
The Efficient Market Hypothesis
If you're so smart, why aren't you rich? Why 90% of fund managers underperform a dart-throwing monkey over 15 years.
The Principal-Agent Problem
When the person you hired has different incentives than you. Holmström's optimal contracts, CEO pay puzzles, and why your real estate agent doesn't care about your last $10,000.
Prediction Markets
Betting as a truth machine. Why Polymarket outpredicts polls, how calibration works, and the math of turning crowds into probabilities.
Hyperbolic Discounting
Why you eat the cake today and plan the diet for Monday. The math of time-inconsistency and why Ulysses tied himself to the mast.
Part XVI: Thinking About Thinking
Cognitive biases, part II — the deeper catalog of how human reasoning systematically fails.
The Conjunction Fallacy
Linda the feminist bank teller. Why 85% of people violate the most basic rule of probability — because stories beat math.
The Availability Heuristic
Why we fear sharks more than vending machines. We judge probability by how easily examples come to mind — and media breaks everything.
Framing Effects
The same surgery, two brochures. "90% survival rate" vs "10% mortality rate" — same math, opposite decisions.
The Sunk Cost Fallacy
Why you stay for the terrible movie. The $15 is gone regardless — but your brain can't let go of the Concorde.
Confirmation Bias
The most dangerous bias of all. Wason's 2-4-6 task proves we seek evidence that confirms, never evidence that disproves.
The Endowment Effect
Why your coffee mug is worth more to you than to anyone else. Ownership inflates value by 2x — and the Coase theorem weeps.
Part XVII: Shapes of Thought
Discrete math and topology — graphs, colors, pigeons, and the most beautiful equation ever written.
The Bridges of Königsberg
The walk that invented graph theory. Euler's 1736 proof that you can't cross all 7 bridges exactly once — and why it matters for Google Maps.
The Four Color Theorem
The map problem that took 124 years and a computer. The first major theorem proved by machine — and the philosophical crisis it caused.
The Pigeonhole Principle
The simplest idea with the deepest consequences. If N+1 pigeons enter N holes, two must share. Trivial to state, impossible to escape.
Ramsey Theory
Why complete disorder is impossible. Among any 6 people, 3 must be mutual friends or mutual strangers — and the numbers only get crazier.
Euler's Identity
The most beautiful equation in mathematics: eiπ + 1 = 0. Five constants, three operations, one equation that unites all of math.
Fibonacci & the Golden Ratio
The numbers that grow like rabbits. Sunflower spirals, the golden angle, and why most "golden ratio in art" claims are debunked.
Part XVIII: Information & Signal
Information theory and collective intelligence — entropy, signals, word frequencies, and the wisdom of crowds.
Shannon Entropy
How to measure surprise. Shannon's 1948 masterpiece founded information theory and put a number on uncertainty itself.
Signal Detection Theory
The math of finding needles in haystacks. ROC curves, d-prime, and why every alarm system embeds a moral judgment.
Zipf's Law
The mysterious law governing words, cities, and wealth. The second-most-common word is exactly half as common as the first — and nobody fully knows why.
The Wisdom of Crowds
When mobs are smarter than experts. Galton's ox-weighing contest, the Ask the Audience lifeline, and when independence breaks everything.
Fermi Estimation
How to guess anything within an order of magnitude. Piano tuners in Chicago, nuclear yields from paper scraps, and the Drake equation.
Publication Bias
The file drawer problem. 20 labs test jelly beans; one finds p<0.05 by chance. That one publishes. Science eats itself.
Part XIX: Fairness & Allocation
Justice by the numbers — dividing cakes, measuring inequality, and the math of public goods.
Fair Division
I cut, you choose: the mathematics of envy-free allocation. From cake cutting to rent splitting, fairness has at least four mathematical definitions.
Pareto Efficiency
When making someone better off means making someone worse off. A concept economists love and critics loathe — because feudalism is Pareto efficient.
The Lorenz Curve & Gini Coefficient
Measuring inequality with one number. Sweden 0.27, US 0.39, South Africa 0.63. But the same Gini can hide very different worlds.
Public Goods & Free Riders
Why nobody wants to pay for the park. The streetlight problem, the free-rider dilemma, and Ostrom's Nobel-winning solution.
Rent-Seeking
The economics of fighting over the pie instead of baking it. Tullock contests, lobbying waste, and why total spending approaches the prize value.
Part XX: The Edge of Knowledge
Meta-mathematics and epistemology — when science examines itself, causation gets formal, and the map meets the territory.
The Replication Crisis
When science can't reproduce itself. Only 36% of psychology studies replicated. The systemic problem that's rewriting how research works.
Causal Inference
The hardest problem in science: what causes what? Rubin's counterfactuals, Pearl's do-calculus, and the 2021 Nobel for natural experiments.
The Map and the Territory
All models are wrong, but some are useful. The capstone essay: Borges's 1:1 map, Goodhart's Law, and why knowing which map to trust is the whole game.
P-Hacking
The dark art of torturing data until it confesses. Garden of forking paths, the green jelly bean, and why "p < 0.05" doesn't mean what you think.
These 111 essays extend Jordan Ellenberg's How Not to Be Wrong: The Power of Mathematical Thinking (2014). Each covers a concept Ellenberg might have written — with original research, working simulations, and hand-crafted illustrations. Built with math, not magic.