The Galton board: order from random bounces
Goals
- Students count paths with Pascal’s triangle and binomial coefficients.
- Students turn path counts into probabilities.
- Students see why many random events give a predictable shape.
Before the film: predict
1,000 balls fall through 10 rows of pegs. At every peg a ball goes left or right with equal chance. Sketch where you think they pile up.
The film
Random Balls Build a Perfect Bell Curve — Hypnotic Galton Board · hypnoticpatterns.doner.cloud/video/random-balls-build-a-perfect-bell-curve
The lesson, minute by minute
| 5 min | Predict: each student sketches the pile; collect two or three different sketches on the board. |
|---|---|
| 3 min | Watch the film; compare it with the sketches. |
| 20 min | Worksheet in pairs, each pair at its level: Starter, Core or Challenge. |
| 10 min | Discuss: the questions below. |
| 7 min | Live demo on the board: tilt the board to 60 % and add rows; check the predictions. |
The worksheet
Starter
A board has 3 rows. Draw every left/right path a ball can take. How many are there?
Answer: 2 × 2 × 2 = 8 paths.
How many of the 8 paths end in each of the 4 bins?
Answer: 1, 3, 3, 1: one path to each edge, three to each middle bin.
What is the chance that a ball lands in one of the two middle bins?
Answer: 6 ÷ 8 = 75 %.
Core
With 10 rows, how many different paths can one ball take?
Answer: 2¹⁰ = 1,024.
How many of them end in the middle bin (exactly 5 bounces to the right)?
Answer: C(10, 5) = 252.
What is the chance of the middle bin?
Answer: 252 ÷ 1,024 ≈ 24.6 %.
What is the chance of one of the two outer bins?
Answer: 2 ÷ 1,024 ≈ 0.2 %.
Out of 1,000 balls, about how many land in bins 4, 5 or 6?
Answer: (210 + 252 + 210) ÷ 1,024 ≈ 65.6 %, so about 656 balls.
Challenge
Every peg now sends 60 % of balls to the right. Which bin of the 10 is most likely, and how likely?
Answer: Bin 6: C(10, 6) · 0.6⁶ · 0.4⁴ ≈ 25.1 %. The centre of the bell moves to 10 × 0.6 = 6.
With an even number of rows n, the middle bin gets C(n, n/2) ÷ 2ⁿ of the balls. From how many rows is that below 10 %?
Answer: From 64 rows: C(64, 32) ÷ 2⁶⁴ ≈ 9.9 %. The bell spreads, so the middle bin gets a smaller share.
Discuss
- One ball tells you almost nothing. Why do 1,000 balls tell you almost everything?
- Where else do you meet a bell curve? (heights, measurement errors, test scores)
- What would change if every peg sent 60 % of balls to the right?
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