NUMB3RS - Season 3 - "Spree, Part II - Daughters" - Galton Board

At each pin on an unbiased Galton board, a falling ball has two options: roll left or roll right. Since these outcomes are equally likely, the probability of taking either path is exactly $P(\text{Left}) = P(\text{Right}) = \frac{1}{2}$.

At the first peg, the probability splits into 1/2 left and 1/2 right. When a ball encounters the next level of pins, these probabilities split again. For example, to reach the far left bin on a board with 2 rows of pins, the ball must roll left twice: $P(\text{Left, Left}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Because the choices are independent and binary, the number of paths to each bin matches the coefficients in Pascal's Triangle!