The Galton board, also known as a quincunx or bean machine, is a device for statistical experiments named after English scientist Sir Francis Galton. It consists of an upright board with evenly spaced nails (or pegs) driven into its upper half, where the nails are arranged in staggered order, and a lower half divided into a number of evenly-spaced rectangular slots. The front of the device is covered with a glass cover to allow viewing of both nails and slots. In the middle of the upper edge, there is a funnel into which balls can be poured, where the diameter of the balls must be much smaller than the distance between the nails. The funnel is located precisely above the central nail of the second row so that each ball, if perfectly centered, would fall vertically and directly onto the uppermost point of this nail's surface (Kozlov and Mitrofanova 2002). The figure above shows a variant of the board in which only the nails that can potentially be hit by a ball dropped from the funnel are included, leading to a triangular array instead of a rectangular one.
Each time a ball hits one of the nails, it can bounce right with probability 
or left with probability 
. For symmetrically placed nails,
. In the idealized model, a ball
passing through 
rows makes 
independent Bernoulli
trials, each with the same right-turn probability 
. Numbering the 
bottom slots from 0 to 
, the ball reaches slot 
iff it takes exactly 
right turns, which occurs with probability
 |
where 
.
The final slot therefore has a binomial distribution.
As the number of independent balls increases, the fraction in each slot approaches its binomial probability
by the weak law of large numbers. For any
fixed 
,
the central limit theorem implies that the
binomial distribution, after centering and
scaling, approaches a normal distribution
as the number 
of rows increases. Thus, with sufficiently many rows, the heap heights approximate
a normal curve once sufficiently many balls have been dropped.
Some care is needed to obtain these idealized results, however, as the actual distribution of balls depends on physical properties of the setup, including the elasticity of the balls (as characterized by their coefficient of restitution), the radius of the nails, and the offsets of the balls over the funnel's opening when they are dropped (Kozlov and Mitrofanova 2002).
