Each ball falling through the pegboard makes one coin flip per row: left or right, 50–50. Where it lands is exactly the number of “rights” in those flips, so the bin heights are the binomial distribution B(rows, ½), gathered one sample at a time. Increase the row count or pile up balls and the pile starts tracing the same bell that the central limit theorem promises — the sort of thing that happens every single time without anybody arranging it.

The dashed outline over the bins is the expected binomial count; the solid bars are what actually fell. They agree better than they have any right to.