Every ball bounces left or right by pure chance — yet the pile always forms a perfect bell curve. This is the Central Limit Theorem: order emerging from randomness, and the reason so much of nature is normally distributed.
A ball has 50/50 odds at every peg, so landing in a given bin follows a binomial distribution — many routes lead to the middle, few to the edges. As the number of balls grows, that binomial tends to the bell-shaped normal curve. This is why heights, measurement errors and sums of tiny random effects all cluster around a mean: the Central Limit Theorem is doing the same averaging you see here.