Much of this website begins with a deceptively simple question and ends with a probability distribution, a model, or an unexpectedly long calculation. This collection brings together recreational mathematics, statistics, Bayesian modelling and puzzles, with routes into their applications in football and space weather.

Probability and statistical thinking

• The birthday problem — why shared birthdays become likely surprisingly quickly.
• Bayes’ theorem: Maths on a Mug 15 — updating probabilities when new evidence arrives.
• An accessible introduction to extreme value theory — the statistics of rare events, including severe space weather.
• Are all World Cup group draws equally likely? — how constraints change probabilities.

Bayesian models and forecasting

My Kalman filter resources and tutorial notes connect Bayes’ theorem to estimating a changing system from noisy observations. Try the bouncing-ball example, then see the same family of ideas used in football team-strength modelling.

For a model-validation example, read how I compared Euro 2020 predictions alongside the final results. A forecast needs to be judged by more than whether one memorable prediction came true.

Recreational maths and useful derivations

• Hilbert’s Hotel and infinity
• The Crawshaw method for quadratic equations
• The cosine rule on a sphere and the spherical sine rule
• The speed needed to complete a loop-the-loop
• The mathematics of the pound coin

Maths on a Mug

The Maths on a Mug collection starts with an equation on a mug and explores the idea behind it. It is a less chronological way into topics ranging from familiar formulae to the mathematics of extremes.

Browse every mathematics article or explore the interactive tools.