Galton Board — Probability Made Visible
The Galton Board (also called the quincunx or bean machine) was invented by Sir Francis Galton around 1873. A ball dropped from the top encounters a triangular array of pegs; at each peg it bounces left or right with probability p and 1−p respectively. After n rows, the ball lands in one of n+1 bins. The resulting histogram — accumulated over thousands of balls — converges to the normal distribution.
The Mathematics
Binomial Distribution
Let X = number of right-turns in n rows with probability p each. Then X ~ Binomial(n, p):
P(X = k) = C(n,k) · pᵏ · (1−p)^{n−k} k = 0, 1, …, n
E[X] = np (mean bin)
Var[X] = np(1−p) σ = √(np(1−p))
Example (n=12, p=0.5): E[X] = 6, σ = √3 ≈ 1.732
Central Limit Theorem Connection
Each peg decision is an independent Bernoulli(p) trial. By the CLT, the normalised sum converges to N(0,1):
(X − np) / √(np(1−p)) → N(0,1) as n → ∞
Local CLT (de Moivre-Laplace): P(X=k) ≈ φ((k−np)/σ) / σ
where φ(z) = (1/√(2π)) e^{−z²/2} is the standard normal PDF.
Berry-Esseen error: |P(X_n ≤ x) − Φ(x)| ≤ C·ρ/(σ³√n)
(guaranteed O(1/√n) convergence for distributions with ρ = E[|X−μ|³] < ∞)
Why the Dashed Curve Fits
The white dashed overlay shows the theoretical Binomial(n, p) PMF scaled to the observed total ball count. As you drop more balls the empirical histogram converges to this curve by the Law of Large Numbers. For 12 rows it closely matches by ~500 balls; for 18 rows you need ~2000 for a smooth bell. Try dragging p away from 0.5 — the theoretical curve shifts, confirming that p alone determines the distribution's location.
Galton's Original Insight — Regression to the Mean
Galton used the board to illustrate regression to the mean: tall parents have tall children, but children cluster around the population mean more than the parents. The spread of each generation follows a normal distribution with the same variance, a phenomenon he termed "reversion." This insight became the foundation of correlation theory and linear regression.
Preset Comparison
| Preset | Rows | p | E[bin] | σ | Pattern |
|---|
| Fair Coin | 12 | 0.50 | 6.00 | 1.73 | Symmetric bell (centre 6) |
| Biased Left | 12 | 0.30 | 3.60 | 1.59 | Bell shifted left |
| Biased Right | 12 | 0.70 | 8.40 | 1.59 | Bell shifted right |
| Deep Board | 16 | 0.50 | 8.00 | 2.00 | Narrower relative bell |
| Slow Motion | 10 | 0.50 | 5.00 | 1.58 | Wider relative bell |
| Loaded Die | 14 | 0.40 | 5.60 | 1.83 | Asymmetric, left-leaning |
Frequently Asked Questions
Why does the Galton Board produce a bell curve?
Each ball makes n independent binary choices (left/right), so the total rights count follows Binomial(n, p). By the CLT the binomial approximates a normal distribution N(np, np(1−p)) for large n. With 12 rows and fair coin (p=0.5): mean=6, σ≈1.73. The pattern becomes visible after ~100 balls and sharp after ~1000.
What happens when the bias probability is not 0.5?
The distribution shifts: p=0.7 puts most balls in the right bins (mean at 0.7n); p=0.3 puts most balls left (mean at 0.3n). The shape remains approximately normal by the CLT but is skewed for very small or very large p, especially with few rows. The theoretical overlay shows the exact Binomial PMF matching observations.
How many balls are needed before the pattern is clear?
With 12 rows (~100–200 balls to see the rough shape, ~1000 for close match). Fewer rows need proportionally more balls for a smooth histogram (less granularity). The Berry-Esseen theorem shows convergence rate is O(1/√n_balls). Use the "deep board" preset (16 rows, fast rate) to accumulate thousands quickly.
Who invented the Galton Board?
Sir Francis Galton invented the board around 1873 to demonstrate the normal distribution physically. He called it the quincunx. Real Galton boards are still manufactured today. Galton used it to illustrate regression to the mean in heredity, which led directly to his development of statistical correlation. Replica boards are popular demonstrations of the CLT in physics and mathematics education.
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