Why is the bell curve everywhere? If you sum many independent random ±1 steps, does the result always become Gaussian — and how fast?
▶ Launch the interactive simulationDrop thousands of balls through n rows of pegs (each a ±1 Bernoulli step with probability p); the count of rights is a sum of n independent steps, binned into a histogram with the exact Gaussian N(np, np(1−p)) overlaid. A separate exact experiment plots the maximum pointwise gap Δ(n) between the binomial and its Gaussian limit, log-log, across n = 4…256.
the CLT: the histogram fills the bell with measured mean → np and standard deviation → √(np(1−p)); and the convergence RATE — Δ ∝ n^(−3/2) for a symmetric board (the leading skewness correction vanishes at p=½), slowing to Δ ∝ 1/n once the board is biased (p≠½)