Galton Board — Definition, Formula & Examples

A Galton board is a physical device where balls drop through rows of pegs, bouncing randomly left or right at each peg, and collect in bins at the bottom to form a bell-shaped (normal) distribution.

A Galton board consists of $n$ rows of evenly spaced pegs. A ball striking each peg undergoes an independent Bernoulli trial with probability $p = 0.5$ of deflecting left or right. After $n$ rows, the number of rightward deflections follows a binomial distribution $B(n, 0.5)$ , which approximates a normal distribution for large $n$ by the Central Limit Theorem.

Key Formula

P(k) = \binom{n}{k} \left(\frac{1}{2}\right)^n

Where:

$n$ = Number of rows of pegs (total deflections per ball)
$k$ = Number of rightward deflections (determines which bin the ball lands in)
$\binom{n}{k}$ = Binomial coefficient — the number of distinct paths to bin k

How It Works

Drop a ball at the top of the board. At each row, it hits a peg and goes either left or right with equal probability. After passing through all rows, it lands in one of the bins at the bottom. The middle bins collect the most balls because there are far more paths leading to the center than to the edges. Repeating this with hundreds of balls produces the characteristic bell curve. The number of paths to each bin corresponds exactly to the entries in Pascal's triangle.

Worked Example

Problem: A Galton board has 4 rows of pegs, creating 5 bins (labeled 0 through 4 by number of right deflections). What is the probability a single ball lands in bin 2 (the center)?

Identify values: There are

n = 4

rows and we want

k = 2

rightward deflections.

Compute the binomial coefficient: Count the number of paths that give exactly 2 right turns out of 4.

\binom{4}{2} = \frac{4!}{2!\,2!} = 6

Apply the formula: Each of the

2^4 = 16

paths is equally likely.

P(2) = \frac{6}{16} = \frac{3}{8} = 0.375

Answer: The probability of landing in the center bin is

\frac{3}{8}

, or 37.5%. This is the highest probability among all five bins, which is why the center of a Galton board accumulates the most balls.

Visualization

Why It Matters

The Galton board gives a tangible, visual proof that many independent random events combine to produce a normal distribution. This idea underpins statistics courses, quality control in manufacturing, and any field where measurement errors or natural variation follow a bell curve.

Common Mistakes

Mistake: Assuming each bin is equally likely since each single deflection is 50/50.

Correction: The bins are not equally likely. There are many more paths to the center bins than to the edges. For 4 rows, the center bin has 6 paths while each edge bin has only 1.