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Probability & Statistics

From Galton boards and Buffon's needle to Monte Carlo estimation, random walks and Markov chains, this category lets you watch randomness give rise to order through the law of large numbers and the Central Limit Theorem. Each interactive simulation runs live in your browser, so you can change parameters, draw thousands of samples in seconds and see probability distributions, convergence and variance emerge before your eyes. You will learn how Monte Carlo methods estimate π and integrals, how Bayesian inference updates beliefs from data, and how stochastic processes model diffusion and percolation. These ideas matter because they underpin modern statistics, machine learning, finance, epidemiology and scientific computing — turning abstract theorems into intuition you can genuinely feel and reuse.

6 simulations Canvas 2D · Three.js LLN · CLT · Brownian

Probability Simulations

Click any card to open the simulation in your browser

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New ★★☆ Moderate
Benford's Law
Explore the surprising frequency of leading digits in real-world data. Compare empirical digit distributions against the theoretical log distribution and run a chi-squared fraud-detection test.
Logarithmic Fraud Detection Chi-squared Canvas 2D
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New ★★★★ Advanced
Hidden Markov Model
Watch a hidden Markov model emit observations from latent states, then recover the most likely state path with the Viterbi algorithm and infer stat…
HMM Markov forward algorithm
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New ★★☆ Moderate
Reservoir Sampling
Pick k uniform samples from a stream of unknown length in one pass: item i replaces a reservoir slot with probability k/i.
reservoir sampling streaming Algorithm R
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New ★★★ Advanced
Dynamic Time Warping
Dynamic time warping aligns two time series that vary in speed by warping the time axis.
DTW time series dynamic programming
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★★☆ Moderate
Secretary Problem
Reject the first ~37% of candidates, then take the first better than all seen. Success peaks at N/e and converges to 1/e ≈ 0.368.
Optimal Stopping 1/e Rule Canvas 2D
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★★☆ Moderate
Buffon's Needle
Drop needles on a lined floor and estimate π ≈ 2L·N / (d·crossings). Watch the Monte Carlo estimate converge by the law of large numbers.
Monte Carlo Probability Canvas 2D
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★★☆ Moderate
Coupon Collector's Problem
How many random draws to collect all N coupons? Compare the empirical average to E[T]=N·H_N and the N·ln N+γN approximation.
Harmonic Numbers Expectation Canvas 2D
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Popular ★☆☆ Easy
Galton Board
Balls fall through a triangular peg array. Each peg is a Bernoulli trial — watch the binomial distribution converge to a Gaussian as ball count grows.
Binomial CLT Canvas 2D
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★☆☆ Easy
Monte Carlo Pi
Scatter random points in a unit square. Points inside the inscribed circle: π ≈ 4 × hits / total. Live convergence plot shows √N error decay.
Monte Carlo Estimation Canvas 2D
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★★☆ Moderate
SIR Epidemic Model
Stochastic SIR: each contact is a Bernoulli trial with transmission probability β. Realise different epidemic curves from the same parameters — variance in action.
Stochastic Bernoulli Three.js
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★★☆ Moderate
Random Walk & Diffusion
2D random walk: mean squared displacement grows as ⟨r²⟩ = 4Dt. Toggle correlated (Lévy) walk to see anomalous diffusion with fatter tails.
Brownian Diffusion Lévy Walk
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★★☆ Moderate
Genetic Drift
Wright–Fisher sampling: each generation chooses N alleles at random from the parent pool. Watch small populations fix alleles by chance, not selection.
Sampling Gene Frequency Canvas 2D
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★★★ Advanced
Percolation
Open each site with probability p. Near the critical threshold p_c ≈ 0.5927, spanning clusters fractal geometry and power-law cluster-size distributions emerge.
Phase Transition Fractal Critical
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★☆☆ Beginner
Monte Carlo π
Estimate π by throwing random darts at a unit square with an inscribed circle. Watch the ratio converge to π/4 on a live convergence chart.
Monte Carlo Random Sampling Canvas 2D
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★★☆ Moderate New
Probability Distributions
Explore Normal, Binomial, Poisson, Exponential, and Uniform distributions interactively. Toggle between PDF and CDF, overlay two distributions, and read off P(x ≤ cursor) as you hover.
PDF / CDF Statistics Canvas 2D
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New ★★☆ Moderate
Central Limit Theorem
Draw repeated samples from Uniform, Exponential, Bimodal or Poisson distributions and watch the histogram of sample means converge to a normal bell curve.
Statistics Sampling Normal Distribution
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New ★★★ Advanced
Bayesian Inference
Watch a Beta-Binomial posterior update in real time as you add observations. Choose from four scenarios (biased coin, medical test, rate estimation, custom), adjust the prior and see the 95% credible interval shift with each new data point.
Bayes Theorem Beta Distribution Posterior
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★★☆ Moderate New
Markov Chains
Animated state transition diagram with random walk. Compare empirical visit frequencies against the theoretical stationary distribution. Presets: weather, PageRank, gambler's ruin.
Transition Matrix Stationary Distribution Canvas 2D
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New ★☆☆ Beginner
Linear Regression — OLS
Click to add data points, watch the least-squares line update live. Explore slope, intercept, R² and residuals.
OLS R Squared Statistics
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★★☆ Moderate
Student's t-Test
Interactive one-sample, two-sample and paired t-tests with live scatter plots and t-distribution visualisation. p-value, Cohen's d, confidence intervals and statistical power computed live.
Hypothesis Testing p-value Effect Size Canvas 2D
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★★☆ Moderate New
Lévy Flight
Anomalous diffusion simulator using the Mantegna algorithm for symmetric α-stable distributions. Tune the stability index α from Brownian (α=2) to Cauchy (α→1). Live log-log step histogram reveals the characteristic power-law tail.
Stable Distribution Power Law Mantegna Canvas 2D
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New ★☆☆ Beginner
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New ★★☆ Moderate
Regression to the Mean
Explore why extreme performances tend to be followed by more average ones. Bivariate normal scatter plot with correlation slider — see the sports curse and medical placebo effect explained statistically.
Bivariate Normal Correlation Sports Curse Canvas 2D
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New ★★★ Advanced
Stochastic Processes
Random walks, Brownian motion, geometric Brownian motion, Poisson processes and the central limit theorem.
random walk Brownian motion geometric Brownian
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New ★★★ Advanced
Wiener Process & Stochastic Calculus
Simulate Brownian motion, Ornstein-Uhlenbeck, and Geometric Brownian Motion — the mathematical foundation of financial modelling. Euler-Maruyama discretisation with up to 30 live paths.
Brownian Motion GBM Black-Scholes Canvas 2D
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New ★★☆ Moderate
Gambler's Ruin
Simulate the classic gambler starting with k dollars betting against a house of n dollars. Discover why the house always wins — and find the Kelly optimal bet fraction.
Ruin Probability Random Walk Kelly Criterion Canvas 2D
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New ★★★ Advanced
Monte Carlo Importance Sampling
Estimate rare-event probabilities efficiently. Importance sampling: E[f(X)] = E_q[f(X)w(X)] where w ...
Monte Carlo Rare Events Variance Reduction
Galton Board
Balls cascade through a triangular peg array — each peg is a Bernoulli trial. Watch the binomial histogram converge to the normal bell curve as ball count grows.
Binomial Distribution CLT Canvas 2D

Key Concepts

Probabilistic ideas that appear across many simulations

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Law of Large Numbers
As N → ∞, sample mean converges to population mean. The fundamental justification for why simulation results are trustworthy statistics.
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Central Limit Theorem
Sum of N independent random variables (finite variance) tends to a Gaussian regardless of individual distributions. Explains ubiquity of the bell curve.
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Brownian Motion
Continuous-time limit of a random walk. Mean squared displacement ⟨r²⟩ = 2dDt in d dimensions. Underpins diffusion and stochastic calculus.
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Monte Carlo Methods
Approximate integrals and probabilities by random sampling. Error scales as 1/√N independent of dimension — invaluable for high-dimensional integration.

Distributions in These Simulations

Approximate prevalence by simulation count

Gaussian / Normal
85%
Uniform
60%
Binomial
45%
Poisson
30%
Power-law / Pareto
20%

Learning Resources

Articles and tutorials about the algorithms in this category

Article Bayes' Theorem in Practice Prior, likelihood, posterior. Bayesian updating, conjugate priors and MCMC sampling with Metropolis-Hastings. → Article Monte Carlo Methods Estimating π, integration by random sampling, variance reduction techniques and quasi-Monte Carlo sequences. → Article Markov Chains & Random Walks Transition matrices, stationary distributions, PageRank and the gambler's ruin problem. →
All articles →

About Probability & Statistics Simulations

Random walks, Monte Carlo, distributions, and sampling — made visible

Probability and statistics simulations make randomness concrete and measurable. Random-walk simulations plot thousands of independent Brownian-motion paths and watch the average displacement grow as √t, directly verifying Einstein's diffusion equation. Monte Carlo circle-area estimators converge toward π with each random point dropped, demonstrating the law of large numbers in real time.

Central-limit-theorem visualisers show how the sum of any bounded independent random variables converges to a Gaussian distribution as sample size grows. Dice-roll and coin-toss simulators compute empirical distributions across millions of trials in seconds. These interactive experiments build statistical intuition for concepts that underlie every branch of science, engineering, and data analysis — from hypothesis testing to Bayesian inference.

Each simulation in this category is built with accuracy and interactivity in mind. The underlying mathematical models are the same ones used in academic research and professional engineering — just made accessible through a web browser. Changing parameters in real time and observing the results is one of the most effective ways to build intuition for complex scientific and engineering concepts.

Frequently Asked Questions

Common questions about this simulation category

Do these simulations require installation?
No. Every simulation runs entirely in your web browser using WebGL and Canvas 2D. Nothing to install or download — open the page and the simulation starts immediately.
Can I use these simulations for teaching?
Yes — all simulations are designed to be educational and run without an account or login. They are widely used in university lectures, high-school science classes, and self-directed learning. Embed them via iframe or link directly.
What devices do the simulations support?
All simulations work on desktop browsers (Chrome, Firefox, Edge, Safari). Many work on mobile and tablets too, though some physics-heavy simulations benefit from the GPU performance of a desktop or laptop.

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Every Probability & Statistics simulation on this page runs free in your browser, so you can learn Probability & Statistics online without any installation. Use each interactive Probability & Statistics model to experiment with random walks, Monte Carlo sampling, the Central Limit Theorem and Bayesian inference, then apply the same techniques to real-world problems such as financial risk modelling, A/B testing, quality control and epidemic forecasting. Whether you are a student, teacher or data analyst, these visual experiments build lasting statistical intuition you can carry into research, engineering and everyday decision making.