Fundamentals
0 guidesThe mental models - expected value, probability, Bayes, and the maths that underpins clear thinking.
All fundamentalsA field guide to thinking under uncertainty
Think in probabilities, not certainties.
Most bad decisions are not bad luck - they are clear thinking gone wrong under uncertainty. This is a growing library on expected value, Bayes and the biases that trip us up, written in plain English with the maths shown, not hidden.
E[X] = Σ xi · p(xi)
Expected value is just every outcome, weighted by how likely it is. It sounds simple, and it is - but almost nobody actually thinks this way. We chase vivid risks, anchor on first numbers, and mistake a confident story for a likely one. The fix is a handful of mental models you can learn in an afternoon and use for the rest of your life.
Four ways in
The library splits into four lenses. Start anywhere - each essay stands on its own.
Cognitive Biases
0 guidesThe systematic errors that distort how we read probability - and the habits that fix them.
All cognitive biasesStrategy
0 guidesTurning probability into action - betting, sizing, decision trees and calibration.
All strategyInvesting
0 guidesPosition sizing, risk, and thinking probabilistically about markets and money.
- Position sizing with an edge
- Confirmation bias in investing
- How prediction markets work
- Why active traders underperform
If you read three things
The essays the rest of the site builds on. Read these in order and the others click into place.
01 Start here Expected value, explained The single most useful idea on the site: weigh each outcome by how likely it is. Once it clicks, you spot good bets - in money, in careers, in everyday life. 02 Updating Bayes' theorem, the right way How to revise a belief when new evidence arrives - without over-reacting to the latest thing or stubbornly ignoring it. Five worked examples, no jargon. 03 Sizing How much to bet You have an edge. Now what? The Kelly criterion gives the mathematically optimal stake - and why half of it is usually the wiser default.
The fun part
Results that feel wrong
Probability is full of answers that fight your intuition - and are correct anyway. These are the ones people argue about.
⅔ vs ½ Always switch doors Switching doubles your odds. Almost nobody believes it the first time. 99% accurate A test that is wrong half the time A 99%-accurate screen for a rare condition is more likely wrong than right. In court How courts misread the odds Confusing two different probabilities has convicted innocent people. Hidden data The data you never see The planes that came back told the wrong story. So do most success stories. Never 'due' Nothing is ever due A fair coin that lands heads ten times is still 50/50 on the eleventh. Extremes fade Why extremes never last The rookie of the year slumps next season - and it is not a curse.
Don't just read it - run the numbers
Free calculators built into the guides. Change the inputs and watch the answer move.
Calculator Expected value Enter outcomes and probabilities; see the EV and whether the bet is worth taking. Calculator Kelly stake Your edge and the odds in, the optimal (and half-Kelly) stake out. Calculator Bayes / false positives Set a test accuracy and a base rate; watch the real chance you are positive.
New here?
There's a reading order
Start Here lays out a guided path - the one concept everything builds on, then the biases quietly shaping your decisions, then where to apply it all.
New essays, occasionally
One short email when something new goes up - probability, decision-making, and the odd counterintuitive result. No noise.