Galton Board
A Galton board is a vertical board consisting of several rows of pegs. A number of
beads are released from the top of the board, and these fall and make their way to
the slots at the bottom of the board, bouncing either left or right as they hit the pegs
along the way.
 The random variables in this case are the positions that the beads land in, or
more precisely, the number (or index) of the slot they fall into.
 The random variables are independent because they have no impact on one
another. In other words, if we know that a certain bead landed in a certain slot,
this gives us no information about where the next bead is going to land. The
probability distribution is the same for each bead. This is why the random
variables are independent and identically distributed (i.i.d.).
 The probability distribution for each random variable is a binomial
distribution. If we denote the number of rows of pegs by n, there will be n
Bernoulli trials which are the n bounces each bead undergoes. We can assume
that a success in each trial is the event that the bead bounces towards the right.
If a bead makes k bounces to the right, then it will eventually land in the kth
slot counting from the left. Let the leftmost bin be the 0th bin and the random
variable X be the index of the slot that a bead lands in. The probability that a
bead falls into the kth slot counting from the left is:
𝑛
𝑃(𝑋 = 𝑘) = ( ) 𝑝𝑘 𝑞 𝑛−𝑘
𝑘
Galton Board
where p is the probability of success (probability of bouncing to the right);
and q = 1 – p (the probability of bouncing to the left).
In an unbiased, level Galton board, p = q = 0.5. The probability that a bead falls into
the kth slot thus simplifies into:
𝑛
𝑃(𝑋 = 𝑘) = ( ) 0.5𝑛
𝑘
 The sample size in a Galton board is simply the number of beads in the board.
 According to the de Moivre - Laplace theorem, the binomial distribution
approximates the normal (or Gaussian) distribution when the number of trials
n is large. This is the case in a Galton board with a large number of beads and
rows. A normal distribution has a bell-shaped curve, which explains the bellshaped arrangement of beads in the end.