Predictability arising from randomness? What is this, did I somehow reverse-engineer Desmos's RNG, or is this real?
While reverse-engineering RNG would have been really cool, no, I didn't have to do that for this graph—all I needed was statistics.
Just like the physical variant shown to the right, this digital Galton board (also called a quincunx or bean-machine) demonstrates
two statistical phenomenon: regression to the mean and central limit theorem.
First, I'll explain regression to the mean—the balls are less likely to bounce towards the outside since there are actually less
paths that take the balls there. Consider a 4-tiered pegboard like the default configuration for this graph. To reach the leftmost slot, the
ONLY way to get there is to bounce left four times, which has a 1 in 16 probability if bouncing left is equally as likely
as bouncing right. By contrast, there are 6 unique ways to reach the middle slot (can you count them all?)—a 6 in 16 probability.
As a result, the balls are biased to be closer to the center.
Secondly, central limit theorem. On average, taking many independent random samplings (most staticians agree on 30+)
from any distribution will approximate a normal distribution. This is applicable to the Galton Board, since the path
of each ball is a random sampling of some distribution, and we have a large number of balls to approximate with.
Both my digital version (where a ball bouncing off a peg is modeled with a binomial distribution) AND the physical
board (where balls collide with pegs and each other) produce similar bell curves.
This graph won me a Reddit platinum award on r/desmos.