Theory says the landing slot is Binomial(n, p): mean = n·p, standard deviation = √(n·p·(1−p)). The two columns should converge as beads pile up.
A bead hits a peg and goes left or right — a coin flip. It does that n times on the way down, then lands in a slot. Its slot number is simply how many times it went right. So the pile at the bottom isn't a picture of luck; it's a picture of the binomial distribution:
There is exactly one path to the far-left slot (left, left, left, …) and exactly one to the far right. But there are 924 different paths into the middle slot of a 12-row board. The middle isn't preferred — it's just enormously more reachable.
Turn the rows up to 22 and the jagged binomial melts into the smooth Gaussian curve drawn over it. That's the Central Limit Theorem: add up enough independent random nudges — any kind of nudges, they don't have to be coin flips — and the total is always approximately normal. It's the reason the bell curve turns up in heights, measurement error, test scores, and noise in your headphones. Nature adds a lot of small things together.
Push chance of going right to 70%. The pile slides right and gets narrower — a biased coin is a more predictable coin. Maximum uncertainty, and the widest pile, sits exactly at 50/50.
Sir Francis Galton built the original in the 1870s and called it the quincunx. He used it to argue about heredity; it has outlived the argument.