Interactive visualizations of probability concepts — distributions, laws, paradoxes, puzzles.https://probability.app/https://probability.app/distributions/normal/https://probability.app/distributions/normal/Tune μ and σ; watch the bell shift and stretch against samples drawn live.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/central-limit/https://probability.app/laws/central-limit/Average almost anything enough times and it starts looking Gaussian.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/large-numbers/https://probability.app/laws/large-numbers/Running averages settle. Watch 20 coin-flip runs converge to the same value.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/paradoxes/monty-hall/https://probability.app/paradoxes/monty-hall/Switch or stay? Simulate the game and watch win rates diverge.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/paradoxes/base-rate/https://probability.app/paradoxes/base-rate/A 99% accurate test for a rare disease — what does a positive really mean?Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/birthday/https://probability.app/puzzles/birthday/How many people before a shared birthday is more likely than not?Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/poisson/https://probability.app/distributions/poisson/Count rare events with λ — and watch the bars converge to a Normal as λ grows.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/binomial/https://probability.app/distributions/binomial/n coins, probability p — see the discrete count distribution shift and stretch toward Gaussian.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/exponential/https://probability.app/distributions/exponential/Waiting times between Poisson events — and the only continuous memoryless distribution.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/beta/https://probability.app/distributions/beta/A distribution over probabilities. The Bayesian conjugate prior for coin flips.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/cauchy/https://probability.app/distributions/cauchy/Looks like a Normal but has no mean — the running average never settles.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/bayes/https://probability.app/laws/bayes/Update beliefs when evidence arrives. Area diagram makes posterior probability visible.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/random-walk/https://probability.app/laws/random-walk/Each step is ±1. Spread grows as √t — not t. See what bias does to the drift.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/paradoxes/simpsons/https://probability.app/paradoxes/simpsons/A trend holds in every subgroup but reverses when pooled. Adjust case mixes to see it flip.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/paradoxes/gamblers-fallacy/https://probability.app/paradoxes/gamblers-fallacy/After five heads, tails must be due — right? Running proportions expose the error.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/coupon-collector/https://probability.app/puzzles/coupon-collector/How many boxes to complete the set? E[T] = n·Hₙ — surprisingly large.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/buffon/https://probability.app/puzzles/buffon/Drop needles on lined paper to estimate π. Watch the estimate converge.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/gamblers-ruin/https://probability.app/puzzles/gamblers-ruin/Two players bet until one goes bankrupt. Starting stake and bias determine the odds.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/secretary/https://probability.app/puzzles/secretary/Reject the first 37%, hire the next best. Optimal stopping at 1/e ≈ 37%.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/geometric/https://probability.app/distributions/geometric/Count flips until the first head. The only discrete memoryless distribution.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/log-normal/https://probability.app/distributions/log-normal/Multiply independent shocks and the product goes log-normal. Income, wealth, viral spreads.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/uniform/https://probability.app/distributions/uniform/Every value equally likely — the flattest distribution. Yet sums of uniforms still go Gaussian.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/gamma/https://probability.app/distributions/gamma/Waiting time for k Poisson events. Shape k controls skew; rate λ controls scale.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/regression-to-mean/https://probability.app/laws/regression-to-mean/Extreme values pull back toward average — not because of any force, but pure probability.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/markov-chains/https://probability.app/laws/markov-chains/Wander between states with fixed transition probabilities. Long-run averages stabilize.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/paradoxes/st-petersburg/https://probability.app/paradoxes/st-petersburg/A game with infinite expected value — yet no one would pay much to play. Expected value breaks down.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/paradoxes/two-envelopes/https://probability.app/paradoxes/two-envelopes/A compelling argument says switching always wins. Simulation shows both strategies tie at 50%.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/derangements/https://probability.app/puzzles/derangements/How likely that no hat returns to its owner? Approaches 1/e ≈ 36.8% for any n ≥ 6.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/puzzles/ballot/https://probability.app/puzzles/ballot/A gets a votes, B gets b. P(A strictly ahead throughout) = (a−b)/(a+b). Proved by reflection.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/bernoulli/https://probability.app/distributions/bernoulli/One flip, two outcomes. The atom every other discrete distribution is built from.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/negative-binomial/https://probability.app/distributions/negative-binomial/Flip until the r-th success. Dispersion that Poisson can’t match.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/hypergeometric/https://probability.app/distributions/hypergeometric/Sampling without replacement. Binomial in the limit as the population grows.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/students-t/https://probability.app/distributions/students-t/Heavier tails than Normal at small ν. Shrinks to a Gaussian as degrees of freedom grow.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/chi-squared/https://probability.app/distributions/chi-squared/Sum of k squared standard normals. The engine of variance tests.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/f-distribution/https://probability.app/distributions/f-distribution/Ratio of two chi-squareds. The ANOVA distribution.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/weibull/https://probability.app/distributions/weibull/Failure times. Shape k flips the hazard rate from falling to rising.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/pareto/https://probability.app/distributions/pareto/Power-law tails. The 80/20 rule lived on a log-log axis.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/laplace/https://probability.app/distributions/laplace/Two exponentials back-to-back. The prior behind L1 regularization.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/logistic/https://probability.app/distributions/logistic/The CDF everyone secretly uses. Lighter centre, heavier tails than Normal.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/gumbel/https://probability.app/distributions/gumbel/The distribution of maxima. Extreme-value theory in one curve.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/triangular/https://probability.app/distributions/triangular/Min, mode, max — a cheap stand-in when that is all you know.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/distributions/gaussian-mixture/https://probability.app/distributions/gaussian-mixture/Two bells, one weight. Unmixes into components you can steer by hand.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/markov-inequality/https://probability.app/laws/markov-inequality/P(X ≥ a) ≤ E[X]/a. The simplest tail bound, often the loosest.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/chebyshev/https://probability.app/laws/chebyshev/Tail probability bounded by 1/k². Drag k and watch how loose it really is.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/jensen/https://probability.app/laws/jensen/For convex f: E[f(X)] ≥ f(E[X]). Watch the gap grow as you bend the curve.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/hoeffding/https://probability.app/laws/hoeffding/Concentration of bounded random variables. The bound tightens as n grows.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/bootstrap/https://probability.app/laws/bootstrap/Resample from your sample. Standard errors without any formula.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/confidence-intervals/https://probability.app/laws/confidence-intervals/100 experiments, roughly 95 ribbons cover μ. Fixes the usual misreading.Sun, 19 Apr 2026 00:00:00 GMThttps://probability.app/laws/hypothesis-testing/https://probability.app/laws/hypothesis-testing/Null distribution with a sliding observed statistic. P-values as tail areas.Sun, 19 Apr 2026 00:00:00 GMT