Deep-dive articles on probability theory, mathematical reasoning, and the statistics behind the tools we build.
Most probability tools fall into two traps: casino gloss or 1998 university pages. We wanted a third thing — a tool that takes probability seriously as mathematics.
The 2.70% house edge and 5.26% house edge both sound small. Across thousands of spins the difference is anything but — here is the exact math.
Ask most people how many people fit in a room before two share a birthday. They say 183. The correct answer is 23. Here is the exact math explaining why.
Playing randomly costs you 2–4%. Playing perfect basic strategy compresses the house edge to 0.5%. Every hit, stand, double, and split has a calculable expected value.
Some probability problems are analytically intractable. Monte Carlo simulation solves them by substitution: simulate thousands of trials and count the outcomes.
Flip a coin 10 times and you might get 70% heads. Flip it 1,000,000 times and you will be within 0.05% of 50%. The law of large numbers guarantees this convergence.
In a 52-card deck there are 2,598,960 distinct five-card hands. Every poker probability begins with this number. Here is the complete reference with odds, outs, and pot odds.
Heights, measurement errors, stock returns, IQ scores. Why does the bell curve keep appearing? The Central Limit Theorem is the answer — and it is arguably the most important theorem in statistics.
You test positive for a rare disease. The test is 99% accurate. How worried should you be? In many real scenarios, the answer is: not very. Here is the exact math.
Expected value is the average outcome of a random process over many repetitions. It is arguably the single most useful concept in decision-making under uncertainty.
Standard deviation is not just a formula — it is a measure of how wrong your average is. Understanding it changes how you read every statistic you encounter.
In 1898, a statistician counted soldiers kicked to death by horses. The data followed a distribution that now models everything from earthquakes to Amazon server requests.
A gambler with finite wealth faces a casino with infinite wealth. Even with a fair coin, the gambler is mathematically certain to go broke. The proof is exact and merciless.
A 95% confidence interval does not mean a 95% chance the true value lies inside it. This subtle distinction matters enormously — and almost everyone gets it wrong.
The 0.05 significance threshold has governed scientific publishing for 80 years. It was never meant for that purpose — and the resulting crisis is reshaping statistics.
Francis Galton discovered it in pea plants. It explains why the Sports Illustrated cover curse is real, why bad students improve after punishment, and why no hot streak lasts.
A random walk is the sum of random steps. It describes particle diffusion, genetic drift, and — according to the Efficient Market Hypothesis — stock price movements.
In the 1980s and 90s, a rotating group of MIT students and alumni systematically extracted tens of millions of dollars from Las Vegas casinos using mathematics and a few acting lessons.
Romanian economist Stefan Mandel won the lottery 14 times across three countries. His method was mathematically sound, completely legal, and has since been made impossible to replicate.
In July 1891, Charles Wells arrived at the Monte Carlo Casino with £4,000 and left with £40,000 after winning 23 out of 30 consecutive spins. The math says it was possible. The casinos said it was impossible.