Born to visualize regression
Sir Francis Galton built the “bean machine” in the 1880s to show how random variation regresses toward the mean—literally dropping beads to form a bell curve.
Origin story
Interactive Galton Board Probability Simulator
Run the experiment
How to use the simulator
- Set the number of rows.
- Set the constant probability p of a right bounce.
- Choose the number of balls in the sample.
- Select Drop balls to run the experiment.
- Compare observed results with the exact binomial probabilities in the histogram and table.
Advanced settings
Animated mode shows each path; instant mode skips motion. In step mode, start a step ball and use Next step for one bounce at a time.
Idle
No target
Scrub the bias slider, choose a preset, or switch to instant fill for large experiments.
Balls0
Bins11
Mean (bin)—
SD (bins)—
Expected Mean—Δ —
Expected SD—Δ —
In Flight0Remaining —
Fit RMSE—χ² —
| Bin k | Observed | Observed % | Theoretical probability | Expected | Difference |
|---|
Method and limitations: each bounce is an independent Bernoulli trial with constant right probability p. Animation is not collision physics, balls do not affect one another, and Reset starts a new random sample. Real boards and pachinko machines can differ because geometry, friction, collisions, tilt, launch conditions, and peg imperfections affect motion.
Tip: Increase rows to see the distribution approach a bell curve. Adjust bias to skew left/right.
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How the Galton board model works
This interactive Pachinko / Galton board (also called a bean machine or Quincunx) turns many small, independent decisions into a big picture: a probability distribution. Each ball travels row by row, “bouncing” left or right at every peg. If the chance of going right is p (set with the Bias control), then after n rows the number of right bounces follows a binomial distribution with parameters (n, p). The landing bin corresponds to that count of right steps. With p = 0.5 the distribution is symmetric and, as rows increase, the familiar bell shape appears.
Galton board probability formula
P(X = k) = C(n,k) × pk × (1 − p)n−k.
Here n is the number of rows, k is the landing bin (the number of right bounces), p is the right-bounce probability, and C(n,k) is the number of ways to choose those k right bounces. The mean is np, variance is np(1−p), and standard deviation is √(np(1−p)). The independent, constant-probability model is exactly binomial; a normal curve is only an approximation, generally suitable when np and n(1−p) are both reasonably large.
Worked example: 8 fair rows
For bin 4, C(8,4)(0.5)4(0.5)4 = 70/256 = 27.34%. An edge bin has probability 1/256 = 0.39%. In 1,000 balls, the expected counts are about 273.44 in bin 4 and 3.91 in either edge bin. Middle bins collect more balls because many more paths reach them.
Interpreting results
- Binomial behavior: The probability of landing in bin k (out of 0…n) is C(n, k) · pk · (1−p)n−k. That’s why the theoretical overlay draws a smooth curve right over the live histogram.
- Mean and spread: The expected bin is n·p. Variation is measured by the standard deviation √(n·p·(1−p)). More rows increase spread, but the relative spread shrinks—so the curve looks tighter compared to its center.
- Normal approximation: For moderate/large n, the binomial is well approximated by a normal distribution with mean n·p and variance n·p·(1−p). That’s a hands-on glimpse of the Central Limit Theorem.
- Skew with bias: Set p above 0.5 to skew right; below 0.5 skews left. Extreme biases produce lopsided shapes, great for discussing real-world asymmetry (e.g., wait times, defect counts).
Classroom experiments
The peg layout mirrors Pascal’s triangle: the number of distinct paths to each bin is exactly the binomial coefficient C(n, k). In a STEM classroom, this simulator links combinatorics, probability, and statistics with an engaging visual. In biology, it can illustrate sampling and noise; in engineering, it motivates tolerance stacking and reliability; in computer science, it provides intuition for randomized algorithms.
- Fair vs. biased: Run 500 balls at p = 0.50, then 0.60. Compare the live histogram’s center and shape.
- Law of Large Numbers: Increase the number of drops. Watch sample frequencies settle near theoretical probabilities.
- Estimate p from data: Hide the bias, run a few hundred balls, compute the sample mean, and solve p ≈ mean / n.
- Pascal path counts: For small n, count visible paths to a bin and confirm C(n, k).
Model and references
The model follows the standard NIST binomial distribution; NIST also explains the large-sample normal approximation. Stanford’s probability course provides a concise Galton board demonstration using equal-probability left and right steps. The historical names Galton board, bean machine, and quincunx refer to this probability-demonstration apparatus. Simulation methodology is described beside the results. Last reviewed: July 12, 2026.
Frequently asked questions
What is a Galton board?
A Galton board, bean machine, or quincunx is a peg board where balls make a sequence of left-or-right moves and collect in bins.
Why does a Galton board form a bell curve?
On a fair board, middle bins can be reached by many more left-right paths than edge bins. With enough rows, the exact binomial shape resembles a bell-shaped normal curve.
Is the result binomial or normal?
The idealized model is exactly binomial. A normal curve is only an approximation, usually improving when np and n(1−p) are both reasonably large.
How many bins are produced from n rows?
There are n + 1 bins, numbered 0 through n by the number of right bounces.
What happens when p is not 0.5?
The center moves to np and the distribution becomes asymmetric: values above 0.5 favor higher-numbered bins and values below 0.5 favor lower-numbered bins.
Are bounces independent?
Yes in this mathematical simulator: every bounce is an independent Bernoulli trial with the same right-bounce probability p.
Is this the same as pachinko, Plinko, or a quincunx?
Quincunx and bean machine are names for a Galton board. Pachinko and Plinko use related peg-board layouts, but real games include physical and game-specific effects this model does not calculate.
Why can a physical board differ from the mathematical model?
Real geometry, friction, tilt, collisions between balls, launch conditions, and imperfect pegs can make bounces dependent or give different pegs different probabilities.
5 Fun Facts about Galton Boards & Pachinko
Pascal’s triangle in wood
The number of paths to each bin is exactly C(n,k) from Pascal’s triangle. A Galton board is the triangle turned vertical and sprinkled with gravity.
Combinatorics
Bias reveals skew instantly
Nudge p from 0.50 to 0.55 and the whole histogram slides right. Small biases add up across rows—great for showing how manufacturing tolerances stack.
Skew demo
CLT in motion
As rows increase, the binomial curve hugs a normal curve tighter (Central Limit Theorem). You can watch “approximation” happen live as more balls fall.
Bell curve
Pachinko adds chaos
Real pachinko machines mix fixed pegs with player-controlled launch speed and tilt, turning a Galton board into a game of skill-plus-chaos beloved in Japan.
Arcade twist