A Galton board is a device that demonstrates probability and the binomial distribution. The way the board works can be understood using Pascal's triangle.
In Pascal's triangle, rows of numbers are arranged in the shape of a triangle (each additional row has one additional number than the previous row). The top (or 0th) row is assigned the number 1 and for each new number included in the triangle, its value is the sum of the two numbers touching it from above it, as can be seen here:
Take for example the number 35 in the 7th row, it is the sum of 15 and 20 (the two numbers touching it from above it). Mathematically, each number in Pascal's triangle can be found using the binomial distribution:
where n is the row and k is the placement in the row.
The pegs on a Galton board are arranged in the shape of a triangle. Tiny balls are dropped above the top row. Each time a ball encounters a peg, it can either go left or right (with equal probability). The numbers in Pascal's triangle represent all of the paths a ball can take to reach a given position. For example, the two bottom corners are states only achievable if a ball goes to the left (or right) 13 consecutive times, thus they are labelled with the number 1 because there is only one path that leads to each of these final positions. The positions towards the center have far more possible paths leading to the same final state (1716 paths for the 7th and 8th positions in the 13th row). When a sufficiently large number of balls are dropped, the end result will look like a binomial (or normal) distribution -- from chaos comes order -- as can be seen in the video below.