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Probability Learning Center

Plain-language guides to randomness, probability, and the mistakes people make about chance. Every idea here has a matching tool so you can try it yourself with a coin, a set of dice, or a spinner.

Start here

Understanding Probability: A Beginner's Guide What a probability is, how to read it, and the simple rules for combining events. The foundation for everything else. What Is Randomness? What makes something random, why it rarely looks the way we expect, and where real randomness actually comes from.

The core math

The Law of Large Numbers Explained Why results settle toward their true odds over many trials, and why that never guarantees any single result. Expected Value: Why a 10% Chance Is Not 10% of a Win Odds are not payouts. Why outcomes are all-or-nothing, why the average is often impossible on one try, and why it still decides who wins. The Binomial Distribution Made Simple Counting successes across repeated yes-or-no trials, and why 5 heads in 10 flips is the most likely result. Coin Flips vs Dice Rolls What the two workhorses of chance share, how they differ, and why adding dice makes a bell curve. Is a Coin Flip Truly 50/50? What a fair coin really means, the tiny physical biases research has measured, and why a simulated flip is the fairest of all.

How people misread chance

The Gambler's Fallacy Explained Why no coin, die, or wheel is ever "due", with the Monte Carlo 1913 example and the math of independence. Common Probability Mistakes and Cognitive Biases Base rate neglect, the conjunction fallacy, flipped conditionals, and other traps, each with the mistake and the fix. Why Humans Are Bad at Judging Randomness The pattern-seeking brain, the clustering illusion, and why we expect random to look more mixed than it is.

Casino odds and the house edge

Roulette Odds and Why the House Wins How a single green pocket turns a near-even bet into a steady loss, and why the wheel favors the house on every spin. Craps Odds and Why the House Wins The real odds behind the dice, which bets are close to fair, which are traps, and why time at the table costs you.

Randomness and computers

Random Number Generators Explained How computers produce randomness, why a seed reproduces a sequence, and why security needs a special generator. How Simulations Work The Monte Carlo method: using thousands of random trials to estimate answers a formula cannot easily reach.

Putting it to use

How to Pick a Random Winner Fairly A practical guide to running a raffle or giveaway everyone can trust, and the mistakes that make a draw unfair. Statistics in Everyday Life How probability quietly shapes forecasts, medical tests, insurance, polls, and the apps on your phone.

Try the tools

Coin Flip · Dice Roller · Spinner