Galton board

How it works

At every peg the ball bounces right with probability p. The bin number is the number of right bounces, so after n rows it follows a binomial distribution. Only one path leads to an end bin, while many combinations lead to the middle, which is why balls pile up in the centre. The centre is μ = n·p and the width is σ = √(n·p·(1 − p)). For 14 rows and p = 0.5, μ = 7 and σ ≈ 1.87.

P(k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏμ = n·pσ = √(n·p·(1 − p))

Try it yourself

1. Keep the defaults (14 rows, 50 %, 1,000 balls) and compare “Mean position” and “Standard deviation” with the theoretical 7.00 and 1.87.
2. Set “Probability of going right” to 70 %. Predict where the centre moves: μ = 14 · 0.7 = 9.8, and the distribution narrows slightly to σ ≈ 1.71.
3. Click “Fewer rows” and turn on “Exact distribution”. With only a few rows you can see that the bell curve is just an approximation.
4. Compare 100 and 10,000 balls. More balls smooth the histogram, but only the number of rows and the probability change its shape.

Model limitations

In the model each bounce is independent and has the same probability; it does not simulate real collisions, rebounds or friction. The normal approximation is poor for few rows or for probabilities close to 0 or 100 %.

Related experiments

• Fractals
• Predator and prey
• Double pendulum