Pachinko Probability — Galton Board with Live Distribution

What this shows — from Binomial to Bell Curve

A Pachinko (Galton) board turns many small random choices into a distribution. At each peg the ball “chooses” to go left or right. With a fair board (p = 0.5 to the right), the total number of rights in n rows follows a binomial distribution \( \mathrm{Binomial}(n,p) \). The landing bin is exactly that count. As n grows, the binomial begins to resemble a normal distribution (Central Limit Theorem).

Use the Bias control to set the probability p of going right on each bounce. Values above 0.5 skew the distribution to the right; below 0.5 skews left. The overlay shows the theoretical binomial probabilities for comparison with the live histogram.

The animation uses smooth parabolic arcs between peg centers for clarity and performance (pseudo-physics), while the left/right decision remains random according to your chosen bias.

Learn with a Pachinko (Galton) Board — From Pegs to Probability

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.

What to expect (and why)

• Binomial behavior: The probability of landing in bin k (out of 0…n) is C(n, k) · p^k · (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).

Connections across the curriculum

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.

Ideas for quick activities

• 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).

How the animation works

For clarity and performance, the motion uses pseudo-physics: smooth parabolic arcs between peg centers, while each left/right choice is randomized with probability p. The statistics you see (mean, SD, histogram) always reflect the true underlying binomial model, making this both accurate and easy on the browser.

Tip: Increase Rows to see the bell curve emerge. Then adjust Bias to explore skewed distributions. This “Pachinko probability” simulator is perfect for demonstrations, homework, and quick intuition building.