Nobody can call where a single bead will land, yet a thousand beads are entirely predictable. Each row of pegs is one coin toss, the final slot is the sum of n left-or-right decisions, so the column heights are precisely the binomial coefficients C(n,k) of Pascal’s triangle. With 16 rows the bell is already smooth, just as the Central Limit Theorem promises, and the band μ ± σ holds about 68% of the beads. Push the rightward probability to 0.65 and the peak slides to n·p: still a bell, merely relocated.
Nobody can call where a single bead will land, yet a thousand beads are entirely predictable. Each row of pegs is one coin toss, the final slot is the sum of n left-or-right decisions, so the column heights are precisely the binomial coefficients C(n,k) of Pascal’s triangle. With 16 rows the bell is already smooth, just as the Central Limit Theorem promises, and the band μ ± σ holds about 68% of the beads. Push the rightward probability to 0.65 and the peak slides to n·p: still a bell, merely relocated.
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