Extending Ellenberg

The Missing Chapters

The mathematical ideas Jordan Ellenberg didn't write about — the ones that secretly run your life. Interactive essays with stories, formulas, and simulators.

Companion to "How Not to Be Wrong: The Power of Mathematical Thinking"

17 Chapters · 5 Parts

Choose Your Rabbit Hole

Each essay is self-contained. Start anywhere. They all connect.

Part I
The Bets You Make
Risk, money, and the mathematics of commitment.
Chapter 01
The Kelly Criterion
How Not to Go Broke
Why knowing how much to bet matters more than knowing what to bet on. The formula that connects gambling to information theory — and proves that recklessness is mathematically suboptimal.
Interactive Risk Gambling Investing
Chapter 02
The Ergodicity Problem
Why Economists Get Risk Wrong
The difference between what happens on average across a population and what happens to you over time. One of the most important — and most ignored — ideas in economics.
Economics Probability Risk
Chapter 03
Gambler's Ruin
Why the House Always Wins
The mathematical inevitability of going broke when you play against a larger bankroll. Why casinos don't need to cheat and why persistence isn't always a virtue.
Probability Gambling Random Walks
Chapter 04
Prospect Theory
Why Losing Hurts Twice as Much
Kahneman and Tversky's proof that humans are systematically irrational about risk. The S-curve of value, reference dependence, and why $100 lost stings more than $100 gained.
Interactive Psychology Economics
Part II
When to Decide
Information, belief, and the mathematics of choice.
Chapter 05
Bayes' Theorem
How to Actually Change Your Mind
The 250-year-old formula for updating beliefs with evidence. Why most people are too stubborn or too gullible — and the math that finds the sweet spot.
Interactive Probability Epistemology
Chapter 06
The Secretary Problem
When to Stop Looking and Commit
The mathematics of knowing when you've seen enough. The 37% rule for hiring, dating, apartment hunting, and every other decision where you can't go back.
Decision Theory Optimization Life
Chapter 07
Arrow's Impossibility Theorem
Why Every Election Is Unfair
Kenneth Arrow proved that no voting system can satisfy a handful of reasonable fairness conditions simultaneously. Democracy doesn't have a math problem — it is a math problem.
Social Choice Voting Logic
Part III
Hidden Structure
Patterns hiding in apparent randomness.
Chapter 08
Benford's Law
Numbers Have Fingerprints
Why 1 appears as the first digit 30% of the time, and 9 only 4.6%. The mathematical pattern hidden in tax returns, river lengths, and election results — and how it catches fraud.
Interactive Number Theory Forensics
Chapter 09
Power Laws vs Bell Curves
Extremistan and the Tyranny of the Average
Most people think in normal distributions. The real world runs on power laws — wealth, cities, earthquakes, social media virality. Why the average is a lie.
Interactive Distributions Complexity
Chapter 10
The Birthday Problem
How 23 Strangers Break Your Intuition
In a room of just 23 people, there's a 50% chance two share a birthday. The math of coincidence, combinatorial explosion, and why your intuition about pairs is spectacularly wrong.
Combinatorics Probability Intuition
Chapter 11
Regression to the Mean
The Sports Illustrated Cover Jinx
Why exceptional performance is usually followed by a return to average. The statistical illusion behind the sophomore slump, cursed magazine covers, and the illusion of effective punishment.
Statistics Sports Illusion
Part IV
The Data Lies
Bias, paradox, and the mathematics of illusion.
Chapter 12
Simpson's Paradox
When the Data Lies to Your Face
How aggregating data can reverse the truth. The Berkeley admissions case, kidney stone treatments, and why "the numbers say so" is never the end of the argument.
Statistics Paradox Data
Chapter 13
Survivorship Bias
The Dead Don't Talk
Abraham Wald's bullet holes, startup advice from billionaires, and the systematic error of only studying winners. The most dangerous way to learn from history.
Cognitive Bias Statistics History
Chapter 14
Berkson's Paradox
Why Hot People Seem Like Jerks
When selecting on one variable creates a phantom correlation with another. The dating illusion, hospital data traps, and the hidden math of collider bias.
Paradox Selection Bias Causality
Chapter 15
The Inspection Paradox
Why the Bus Is Always Late
You're more likely to arrive during a long interval than a short one. The length-biased sampling that explains why your friends have more friends than you and class sizes feel inflated.
Probability Sampling Paradox
Part V
Systems That Eat Themselves
Incentives, time, and feedback loops.
Chapter 16
Goodhart's Law
Why Every Metric Eventually Fails
When a measure becomes a target, it ceases to be a good measure. The mathematics of gaming metrics — from Soviet nail factories to modern tech KPIs.
Systems Incentives Economics
Chapter 17
The Lindy Effect
The Older It Is, the Longer It'll Last
Why a book that's been in print for 100 years will likely be in print for another 100. The counterintuitive math of survival applied to ideas, technologies, and careers.
Probability Prediction Philosophy