Mathematics of the Galton Board: From Bernoulli Trials to the Gaussian Distribution

1. What is a Galton Board?

A Galton Board (also known as a quincunx or bean machine) is a physical device invented by Sir Francis Galton to demonstrate fundamental concepts in probability and statistics.

It consists of a vertical board with interleaved rows of pegs. When beads or shot are dropped from a funnel at the top, they bounce through the grid of pegs and collect into a series of vertical bins at the bottom. The resulting accumulation of beads forms a visible curve that illustrates how microscopic random events accumulate into macroscopic probabilistic patterns.

2. The Micro Level: What Each Peg Represents

At the individual peg level, the mechanics represent a simple binary random event:

  • A Single Peg = A Bernoulli Trial
    • When a bead hits a single peg, it must bounce either to the left or to the right.
    • This individual interaction is modeled as a Bernoulli trial, where:
      • Bouncing Right = “Success” (with probability \(p\))
      • Bouncing Left = “Failure” (with probability \(q = 1 - p\))
    • In a standard, symmetric Galton Board, \(p = 0.5\) and \(q = 0.5\).

3. The Macro Level: What All the Pegs Represent

When a bead falls through \(n\) rows of pegs, it undergoes \(n\) independent Bernoulli trials in sequence:

  • The Entire Board = A Binomial Distribution \(B(n, p)\)

    • The path a bead takes is defined by the total number of times it bounces right (\(k\)) out of \(n\) total rows.

    • The probability \(P(k)\) that a bead lands in bin \(k\) is given by the Binomial distribution formula:

      \[P(k) = \binom{n}{k} p^k (1 - p)^{n - k}\]

    • Mean (Center): \(\mu = n \cdot p\)

    • Variance (Spread): \(\sigma^2 = n \cdot p \cdot (1 - p)\)

4. Convergence to a Gaussian Distribution

As beads drop through the machine, the discrete Binomial distribution smoothes out into a continuous bell curve (Gaussian/Normal distribution).

The De Moivre–Laplace Theorem & CLT

This transition is governed by the De Moivre–Laplace Theorem (a special case of the Central Limit Theorem). It states that as the number of trials \(n \to \infty\), the discrete Binomial distribution \(B(n, p)\) converges toward a continuous Normal distribution \(N(\mu, \sigma^2)\):

\[P(k) \approx \frac{1}{\sqrt{2\pi n p(1-p)}} \exp\left( -\frac{(k - np)^2}{2np(1-p)} \right)\]

As thousands of beads fall into the bins at the bottom, their collective heights trace out the Gaussian bell curve in real time.

5. Non-Symmetric Boards: What Happens When \(p \neq 0.5\)?

If the machine is modified—for example, by tilting the board or slanting the pegs so that \(p = 0.7\) and \(q = 0.3\)—it still converges to a Gaussian distribution.

Key Differences with \(p \neq 0.5\):

  1. Shifted Center: The peak moves to the side favored by \(p\). Instead of being centered at the midpoint (\(\mu = 0.5n\)), the peak sits at \(\mu = 0.7n\).
  2. Skewness at Small \(n\): With only a few rows of pegs, the distribution is visibly asymmetric (skewed to the right).

Why Skewness Disappears with Many Rows

The skewness (\(\gamma_1\)) of a Binomial distribution measures its asymmetry:

\[\text{Skewness} = \frac{1 - 2p}{\sqrt{n \cdot p (1 - p)}}\]

  • Because \(n\) is in the denominator, as \(n\) grows larger (\(n \to \infty\)), the skewness approaches 0.
  • Consequently, given enough rows of pegs, the initial asymmetry fades away, and the distribution becomes smooth, symmetrical, and indistinguishable from a Gaussian bell curve around its shifted center \(\mu = n \cdot p\).