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.