October 12th, 2017
Let $B$ be the bucket index that a ball falls into. If you think about it, in a Galton Board, the bucket index is equal to the number of times that a ball goes "right". If it goes right 0 times, it falls into bucket 0. If it goes right 8 times, it falls into bucket 8.
Every ball has 8 chances to go right, and each chance has equal probability of going right or left. If we consider going "right" as a success, we can cast our variable $B$ into the language of a Binomial Random Variable. $B$ equals the number of "successes" in 8 independent trials.
Thus $B$ ~ Bin($n$=8, $p$ = 0.5). We can use the probability mass function of a Binomial to predict the probability of any ball falling into a bucket (which is the same as the fraction of balls that will fall in the bucket).