mathematics

Probability & Randomness

Chance, uncertainty, and the counterintuitive results of probability — from the Monty Hall problem to the law of large numbers.

probabilitystatisticsMonty HallrandomnessGalton boardbirthday paradox

Probability theory is the mathematics of uncertainty. It governs everything from casino games to quantum mechanics, from weather forecasts to machine learning. Yet our intuitions about probability are notoriously unreliable — as the Monty Hall problem and the birthday paradox demonstrate, the correct answer often feels deeply wrong.

These simulations let you explore probability through experimentation rather than formulas. Run thousands of Monty Hall games to see why switching wins 2/3 of the time. Watch balls cascade through a Galton board to form a perfect bell curve. Discover the surprising birthday paradox. Experience the secretary problem's elegant 37% rule.

5 interactive simulations

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Birthday Paradox Simulator

Discover why only 23 people are needed for a 50% chance of a shared birthday — one of probability's most surprising results

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Galton Board (Bean Machine)

Watch balls cascade through pegs to form a perfect bell curve — the physical demonstration of the Central Limit Theorem

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Law of Large Numbers Visualizer

Watch the running average converge to the true mean as samples accumulate — the foundation of all statistical inference

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Monty Hall Problem Simulator

Simulate thousands of Monty Hall games to prove why switching doors wins 2/3 of the time — the most counterintuitive result in probability

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Secretary Problem (Optimal Stopping)

Explore the elegant 37% rule — reject the first third of candidates, then pick the next one who's the best so far