3D Galton Board: Binomial → Normal

3D Galton Board

Peg rows 12

Right-bounce bias p 0.50

Drop rate 3/s

Balls dropped

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In flight

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Bin mean vs R·p

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Bin σ vs √(Rp(1−p))

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How it works

Each falling ball crosses R rows of pegs. At every peg it bounces right with probability p or left with probability 1−p, independently of every previous bounce — exactly a Bernoulli trial. A ball's final bin index is the sum of R independent 0/1 outcomes: a Binomial(R, p) random variable, not a directly-sampled mean like the 2D version. As balls accumulate, the bin heights trace out that binomial distribution, and for large R it looks like a smooth normal bell — the De Moivre–Laplace form of the Central Limit Theorem, demonstrated as a physical sum of coin flips rather than as an average of samples.

bin = Σ Bernoulli(p) over R pegs   (Binomial(R,p))
E[bin] = R·p
SD[bin] = √(R·p·(1−p))

Peg rows R — how many binary bounces each ball accumulates; more rows means a smoother, wider bell.
Right-bounce bias p — the probability of each individual bounce going right; away from 0.5 the binomial is skewed at low R but still pulls toward normal as R grows.
Drop rate — balls released per second; higher rates fill the bins faster without changing the underlying per-ball probabilities.

Real-world relevance: any quantity built from many small, roughly independent contributions — manufacturing tolerances, measurement noise, cumulative sums of yes/no events — tends toward this same bell shape, which is why the normal distribution shows up everywhere in statistical analysis.

Falling ball Peg Bin bar (count)

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3D Galton Board: Binomial Converging to Normal