The Galton Board and the Central Limit Theorem

Drop a ball onto a triangular array of pegs and at every row it bounces left or right, a single coin flip, and after $R$ rows it lands in a bin numbered by how many times it went right. One ball tells you nothing; its path is pure chance. But pour in thousands and a shape emerges that no individual ball intended: a smooth bell, peaked in the middle and tapering at the edges. This is the most tangible demonstration of two of the deepest results in probability. First, the count of rightward bounces is exactly a binomial random variable, so the height of each bin is the binomial probability $\binom{R}{k}p^k(1-p)^{R-k}$, the orange shape the histogram fills out. Second, and more profound, as the number of rows grows that binomial is squeezed into a Gaussian, the famous bell curve, with mean $Rp$ and variance $Rp(1-p)$. That is the central limit theorem: add up many small independent random nudges and their sum is normally distributed, almost regardless of what the individual nudges look like. It is why measurement errors, particle velocities in a gas, and countless other sums in nature are Gaussian. The scene runs the board live, balls cascading into bins; the lower panel normalizes the growing histogram and lays it against both the exact binomial and its Gaussian limit, with a distance meter that shrinks toward zero as more balls fall and the law of large numbers does its work.