Drop a bead onto a triangle of pegs. At every row it bounces left or right — a single coin flip. After a dozen rows its resting bin is the running total of those flips, and where any one bead lands is pure chance. Yet pour thousands of beads through the same pins and an eerie, smooth bell curve assembles itself in the bins below — the very same curve every time. This is the Central Limit Theorem made of falling marbles: sum enough independent coin flips and the histogram of totals is always the normal distribution. Set the rows and the bias, run the physics, and watch the binomial converge to the Gaussian.
A Galton board — Galton's own "quincunx" — is a vertical grid of pegs arranged in a triangle, with collecting bins along the bottom. A bead dropped at the top strikes the first peg and rolls off to one side; it strikes a peg in the next row and again goes left or right; and so on down through n rows. Model each deflection as a coin flip that lands right with probability \(p\) and left with probability \(1-p\). If we let a right count as \(+1\) and a left as \(0\), then the bin the bead lands in is simply \(k=\) the number of rights it took, an integer from \(0\) to \(n\).
Because the flips are independent and identical, the count \(k\) follows the binomial distribution: \(P(k)=\binom{n}{k}p^{k}(1-p)^{n-k}\). The binomial coefficient \(\binom{n}{k}\) is doing the real work — it counts how many distinct left/right paths arrive at bin \(k\). There is exactly one all-left path (bin 0) and one all-right path (bin \(n\)), but there are hundreds of interleaved paths that end up near the middle. The bins aren't equally likely because the middle bins are reachable in far more ways. That combinatorial lopsidedness is the entire source of the mound shape.
Now the miracle. The mean of \(k\) is \(\mu=np\) and its variance is \(\sigma^{2}=np(1-p)\). As the number of rows \(n\) grows, de Moivre and Laplace proved the discrete binomial bars settle onto a continuous curve, \(P(k)\approx \frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\!\big(-\frac{(k-\mu)^{2}}{2\sigma^{2}}\big)\) — the Gaussian, the bell curve. This is the original Central Limit Theorem, and the modern CLT generalizes it enormously: the sum of many independent random contributions tends to a normal distribution regardless of what each individual contribution looks like. The pegs could be biased, the steps could be unequal — sum enough of them and the histogram of totals is still a bell.
This is why the normal distribution is everywhere in nature and measurement. Heights, measurement errors, thermal noise, the totals of many small independent effects — each is a "sum of coin flips" in disguise, and each inherits the bell. The board also makes two subtler lessons tangible. First, bias shifts but does not break the curve: set \(p=0.7\) and the whole mound slides right to centre on \(np\), still bell-shaped, just narrower (variance \(np(1-p)\) is largest at \(p=0.5\)). Second, spread grows like \(\sqrt{n}\), not \(n\): double the rows and the pile widens by only about \(\sqrt2\), because the standard deviation is \(\sqrt{np(1-p)}\). Randomness accumulates, but it accumulates slowly and predictably — a preview of the \(\sqrt n\) law that governs diffusion and random walks.