Question 1

Why does the Galton Board produce a bell curve?

Accepted Answer

Each ball makes n independent binary decisions (left or right at each peg), so the number of right-turns follows a Binomial(n, p) distribution. By the Central Limit Theorem, as n increases the binomial converges to a Normal distribution: N(np, np(1-p)). With n=12 rows and p=0.5, the mean bin is 6 and standard deviation is √3 ≈ 1.73, giving a recognisable bell shape after a few hundred balls.

Question 2

What happens when the bias probability is not 0.5?

Accepted Answer

With p≠0.5 the distribution shifts: biased toward the right (p=0.7) gives mean at bin 0.7n, biased left (p=0.3) gives mean at 0.3n. The bell curve skews and shifts accordingly, though it remains approximately normal by the CLT for large n. The theoretical curve overlay shows the exact expected binomial PMF scaled to the observed ball count.

Question 3

How many balls are needed before the pattern becomes clear?

Accepted Answer

With 12 rows the pattern becomes recognisable after about 100–200 balls, and closely matches the theoretical curve by ~1000 balls. Fewer rows need more balls because the histogram has fewer bins. The Law of Large Numbers guarantees convergence: as total balls n→∞ the empirical histogram converges to the true PMF. The Berry-Esseen theorem quantifies how quickly the binomial approximates the normal.

Question 4

Who invented the Galton Board?

Accepted Answer

Sir Francis Galton devised the board (also called the quincunx or bean machine) around 1873 to demonstrate the normal distribution physically. He used it to visualise how inherited traits regress toward the mean. The device beautifully shows that random processes governed by the Central Limit Theorem inevitably produce normally-distributed outcomes, which Galton connected to his work on regression in heredity.

🧪 Galton Board (Bean Machine) — Binomial Distribution & CLT Simulator

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