Quincunx — Definition, Formula & Examples

A quincunx is a triangular board of evenly spaced pegs where balls are dropped from the top, bouncing left or right at each peg, and collecting in bins at the bottom to form a bell-shaped (normal) distribution.

Also called a Galton board or bean machine, a quincunx is a mechanical device consisting of $n$ rows of staggered pegs through which objects fall under gravity. At each peg, an object has probability $p = 0.5$ of deflecting left or right, so the final bin position follows a binomial distribution $B(n, 0.5)$ , which approximates a normal distribution as $n$ increases.

Key Formula

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

Where:

$n$ = Number of rows of pegs
$k$ = Bin number (0 through n, counting from one side)
$\binom{n}{k}$ = Binomial coefficient — the number of paths leading to bin k

How It Works

A ball enters at the top center and hits the first peg. It bounces either left or right with roughly equal probability. It then hits a peg in the next row and bounces again. After passing through all

n

rows, the ball lands in one of

n + 1

bins at the bottom. Each ball's path is a sequence of independent random events, so the collection of many balls in the bins mirrors the binomial distribution. With enough rows and enough balls, the shape of the bins closely resembles a normal (bell) curve.

Worked Example

Problem: A quincunx has 4 rows of pegs. What is the probability that a single ball lands in the center bin (bin 2)?

Identify values: There are

n = 4

rows, producing 5 bins numbered 0 through 4. The center bin is

k = 2

Apply the formula: Calculate the binomial probability for bin 2.

P(\text{bin } 2) = \binom{4}{2}\left(\frac{1}{2}\right)^4 = 6 \times \frac{1}{16} = \frac{6}{16}

Simplify: Reduce the fraction.

P(\text{bin } 2) = \frac{3}{8} = 0.375

Answer: A ball has a 37.5% chance of landing in the center bin, making it the most likely outcome — consistent with the peak of a bell curve.

Visualization

Why It Matters

The quincunx gives a tangible, visual explanation of why the normal distribution appears so often in nature. In AP Statistics and introductory college courses, it connects the abstract central limit theorem to a physical process you can watch unfold. Simulated quincunx applets are commonly used in classrooms to build intuition about probability distributions before formal proofs.

Common Mistakes

Mistake: Assuming each bin is equally likely.

Correction: The center bins have far more paths leading to them than the edge bins. With 4 rows, bin 2 has 6 paths while bins 0 and 4 each have only 1.