User Guide
Check out the new Galton Board app!
The Galton Board (Simple Edition) with Pascal's triangle is a 6.5 inch by 4.5 inch probability demonstrator providing a visualization of math in motion and the powers of the probabilities and statistics. Printed on the backside of the board is a theoretical investment portfolio histogram, which illustrates the randomness and the probabilities of market returns.
When the Galton Board is turned upside down, the beads flow into the top reservoir. When turn back over and held on a level surface, the 4,280 steel beads and one large golden bead cascade from the reservoir through 14 rows of symmetrically placed hexagons in the Galton Board. When the device is level, beads bounce off of the 105 hexagons with equal probability of moving to the left or right. As the beads settle into one of the 15 bins at the bottom of the board, they accumulate to create a bell-shaped histogram. Flipping the Galton board is like tossing 59,920 coins in about 2 seconds. A bead representing fourteen heads in a row would land in bin #14 and a bead representing no heads (fourteen tails) would land in bin #0.
Printed on the top of the board are formulas for the normal distribution and binomial expansions. Printed on the lower part of the board is the normal distribution or bell curve, as well as the average and standard deviation lines relative to that distribution.
The bell curve, also known as the Gaussian distribution (Carl Friedrich Gauss, 1777-1855), is important in statistics and probability theory. It is used in the natural and social sciences to represent random variables, like the beads in the Galton Board or monthly returns of the stock market. You can also see the Y-axis and X-axis descriptions, and numbered bins with expected percentages and numbers of beads.
Overlaid on the hexagons is Pascal’s triangle (Blaise Pascal, 1623-1662), which is a triangle of numbers that follows the rule of adding the two numbers above to get the number below. The number at each hexagon represents the number of different paths a bead could travel from the top hexagon to that hexagon. It also shows the Fibonacci numbers (Leonardo Fibonacci, 1175-1250), which are the sums of specific diagonals in Pascal’s triangle. Within Pascal’s triangle, mathematical properties and patterns are
numerous. Those include: natural numbers, row totals, powers of 11, powers of 2, figurate numbers, Star of David theorem, and the hockey stick pattern. Other patterns in Pascal’s triangle not identified on this board include prime numbers; square numbers;
binary numbers; Catalan numbers; binomial expansion; fractals; golden ratio; and the Sierpinski triangle.
Among the 4,280 steel beads, there is one larger golden bead, which demonstrates a single random outcome. Shown on top of each bin is the percentage estimates of the probability that a bead will land in that bin. By following the golden bead, you can clearly observe those probabilities with each flip of the Galton Board. With the red investment portfolio histogram on the back, the golden bead can represent the likely range and probabilities of next month’s stock market return. The Galton Board’s probabilities as to which bin the golden bead will land in is a substitute for the prediction of stock market forecasters.
Embedded in this Galton Board are many statistical and mathematical concepts including probability theories, independent identically distributed (iid) random variables, the normal or bell-shaped curve, the Central Limit Theorem (the de Moivre-Laplace theorem), the binomial distribution, Bernoulli (1655-1705) trials, regression to the mean, the law of large numbers, probabilities such as coin flipping and stock market returns, the random walk, the Gambler’s Fallacy, the law of frequency of errors and what Sir Francis Galton referred to as the “law of unreason.”
