an interactive field guide to —
Probability,
made tangible.
Drag the sliders. Watch the samples pile up. Discover why your intuition breaks in all the interesting places — bell curves, Monty Hall, birthday coincidences, and much stranger things.
Start a learning path →
…or pick a flavor ↓
Distributions — Shapes that uncertainty takes
28 pieces →Normal distribution
Tune μ and σ; watch the bell shift and stretch against samples drawn live.
Poisson distribution
Count rare events with λ — and watch the bars converge to a Normal as λ grows.
Binomial distribution
n coins, probability p — see the discrete count distribution shift and stretch toward Gaussian.
Exponential distribution
Waiting times between Poisson events — and the only continuous memoryless distribution.
Beta distribution
A distribution over probabilities. The Bayesian conjugate prior for coin flips.
Cauchy distribution
An undefined mean, a meaningful median, and averages that retain the original heavy tails.
Laws — What happens as you sample more
22 pieces →Central Limit Theorem
See when standardized averages approach a normal shape, and when they do not.
Law of Large Numbers
Compare coin and Cauchy averages, exact error probabilities, and nonmonotone sample paths.
Bayes' theorem
Update beliefs when evidence arrives. Area diagram makes posterior probability visible.
Random walk
Separate drift, spread, and distance; inspect exact endpoint probabilities and seeded paths.
Regression to the mean
Select extreme first readings and compare repeated measurements under an explicit noise model.
Markov chains
Wander between states with fixed transition probabilities. Long-run averages stabilize.
Paradoxes — Where intuition misfires
24 pieces →Monty Hall
Switch or stay? Simulate the game and watch win rates diverge.
Base rate neglect
A 99% accurate test for a rare disease — what does a positive really mean?
Simpson's paradox
A trend holds in every subgroup but reverses when pooled. Adjust case mixes to see it flip.
Gambler's fallacy
After five heads, tails must be due — right? Running proportions expose the error.
St. Petersburg paradox
A game with infinite expected value — yet no one would pay much to play. Expected value breaks down.
Two envelopes paradox
Specify a prior and opening protocol to see why unconditional symmetry does not settle every conditional decision.
Puzzles — Counting problems that bite back
24 pieces →Birthday problem
How many people before a shared birthday is more likely than not?
Coupon collector's problem
How many boxes to complete the set? E[T] = n·Hₙ — surprisingly large.
Buffon's needle
Drop needles on lined paper to estimate π. Watch the estimate converge.
Gambler's ruin
Two players bet until one goes bankrupt. Starting stake and bias determine the odds.
Secretary problem
Reject the first 37%, hire the next best. Optimal stopping at 1/e ≈ 37%.
Derangements
How likely that no hat returns to its owner? The exact probability approaches 1/e ≈ 36.8% as the number of items grows.