The Probable is Usually What Happens

Almost 3 years in the making, The Galton Board is a 7.5” by 4.5” desktop probability machine.

This delightful little device brings to life the statistical concept of normal distribution.

As you rotate the Galton Board on its axis, you set into motion a flow of steel beads that bounce with equal probability to the left or right through several rows of pegs. As the beads accumulate in the bins, they approximate the bell curve, as shown by the yellow line on the front of the Galton board. This hands-on Galton Board allows you to visualize the order embedded in the chaos of randomness.

The Galton Board is math in motion, demonstrating centuries-old mathematical concepts in a modern compact device. It incorporates Sir Francis Galton ’s (1822-1911) illustration of the binomial distribution, which, for a large number of beads approximates the normal distribution. It also superimposes Pascal’s Triangle, which is the triangular array of binomial coefficients (Blaise Pascal, 1623-1662). The binomial coefficient positioned at a given peg counts the number of different paths a bead could take to travel from the top peg to that peg. The Fibonacci numbers can be found as the sums of certain diagonals in the array (Leonardo Fibonacci (1175-1250).

When you rotate the small desktop-sized Galton Board on its axis, 3,000 tiny beads cascade through rows of symmetrically placed pegs. When the device is level, each bead bounces off the pegs with equal probability of moving to the left or right. As the beads settle into the narrow bins at the bottom of the board, they accumulate to approximate the bell curve shown. The Galton Board is reminiscent of Charles and Ray Eames’ groundbreaking 11-foot tall “Probability Machine,” featured at the Eames 1961 Mathematica exhibit. An even larger Eames machine was showcased at IBM’s Pavilion for the 1964 World’s Fair in New York.

SET YOUR MATH BRAIN IN MOTION!

Both the Galton Board and the superimposed Pascal’s Triangle incorporate many related mathematical, statistical and probability concepts. Can you spot them all?

In the Galton Board you may see the Gaussian curve of the normal distribution or bell-shaped curve, the central limit theorem (the de Moivre-Laplace theorem), the binomial distribution (Bernoulli distribution), regression to the mean, probabilities such as coin flipping and stock market returns, the law of frequency of errors, and what Sir Francis Galton referred to as the “law of unreason.”

Within Pascal’s Triangle, you will find mathematical properties and patterns including prime numbers, powers of two, Magic 11’s, Hockey Stick Pattern, triangular numbers, square numbers, binary numbers, Fibonacci’s sequence, Catalan numbers, binomial expansion, fractals, Golden Ratio, and Sierpinkski’s Triangle.

In the Galton Board you may see the Gaussian curve of the normal distribution or bell-shaped curve, the central limit theorem (the de Moivre-Laplace theorem), the binomial distribution (Bernoulli distribution), regression to the mean, probabilities such as coin flipping and stock market returns, the law of frequency of errors, and what Sir Francis Galton referred to as the “law of unreason.”

Within Pascal’s Triangle, you will find mathematical properties and patterns including prime numbers, powers of two, Magic 11’s, Hockey Stick Pattern, triangular numbers, square numbers, binary numbers, Fibonacci’s sequence, Catalan numbers, binomial expansion, fractals, Golden Ratio, and Sierpinkski’s Triangle.

Pascal’s Triangle is a triangle of numbers that follow the rule of adding the two numbers above to get the number below. This pattern can continue endlessly. Blaise Pascal (1623-1662) used the triangle to study probability theory. It also had been studied 500 years previous by Chinese mathematician Yang Hui (1238-1298). The Triangle’s patterns translate to mathematical properties of the binomial coefficients.

The sum of the numbers on the diagonal shown on Pascal’s Triangle match the Fibonacci Numbers. The sequence progresses in this order: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Each number in the sequence is the sum of the previous two numbers. For example: 2+3=5, 3+5=8, 5+8=13, 8+13=21 and so on. Leonardo Fibonacci popularized these numbers in his book Liber Abaci (1202).

As you progress through the Fibonacci Numbers, the ratios of consecutive Fibonacci numbers approach the Golden Ratio of 1.61803398...., but never equal it. For example, 55/34=1.618, and 89/55=1.618, 144/89=1.618. Euclid (325 BC- 270 BC) and other well-known mathematicians studied the properties of the Golden Ratio, including its appearance in dimensions of a regular pentagon and a golden rectangle. Artists and architects, including Dali have proportioned their works to approximate the Golden Ratio, which can also be seen in some patterns in nature, including the spiral arrangement of leaves.

WHENEVER YOU CAN COUNT

THE WISDOM OF THE CROWDS

REVERSION TO THE MEAN

The Galton Board, also known as the Galton Box, is named after Sir Francis Galton (1822-1911). Francis Galton, an English mathematician who was an expert in many scientific fields, created his "Quincunx" machine to demonstrate how a normal distribution is formed through the occurrence of multiple random events.

The Galton Board incorporates many mathematical and statistical concepts, including the normal distribution or bell-shaped curve, the central limit theorem (the de Moivre-Laplace theorem), the binomial distribution, regression to the mean, Pascal’s triangle (the triangle of binomial coefficients), probabilities, powers of two, binary numbers, coin flipping odds, law of frequency of errors, order in apparent chaos, and what Sir Francis Galton referred to as the law of unreason.

Galton wrote in his book Natural Inheritance,

"Order in Apparent Chaos. I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the " Law of Frequency of Error." The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along."

The original Galton Board was built in 1893 and is on display in the Galton Collection, University College London (see image on the left). In the top left corner it states, "Instrument to illustrate the principle of the Law of Error or Dispersion, by Francis Galton FRS. The photo of the Galton Box appears in The History of Statistics by Stephen M. Stigler. For an interesting article on the history of a quincunx instrument see here.

The Quincunx board is used to bring this classic pattern to live, by dropping beads or balls through a series of pegs arrange in a triangular pattern much like Pascale’s Triangle. Balls are dropped onto the first peg and then bounce down to the bottom of the triangle where they collect in little bins. Because the balls have to go through this line of binomial divider, the final number of balls collected at the bins is the same as the total added cumulating of probability of its directional outcome.

In 1964, Charles and Ray Eames built a large working model for the New York World’s Fair exhibit. Their work captures the imagination of many, and bring educational interest of probability to those who saw the display.

Learn More / Additional Resources

Devices

- MathIsFun.com
- The Normal Curve and Galton's Board
- Galton Board simulation (Flash required)
- Making Sense of Randomness
- History of Statistics and the Quincunx
- IFA.tv - A Random Walker
- Gaussian Distrubition Theory
- Institute of Mathematics and its Applications
- The Dance of Chance
- IFA.tv - From Chaos to Order
- Technical University of Crete
- Normal distribution - ADAMED SmartUP
- IFA.tv - Another Random Walker...
- Cambridge Science Festival
- IFA.tv - The Random Walker
- Quincunx LEGO

1964 New York World's Fair

Henry Ford Museum

New York Hall of Science

Boston Museum of Science

- MathIsFun.com
- History and Properties of Pascal's Triangle
- Pascal's Triangle as a Galton Box
- MathWorld
- Wikipedia - Pascal's Triangle
- Pascal's Triangle into Sierpinski Triangle
- Binomial Theorem from Pascal's Triangle
- The Weirdness of Pascal's Triangle
- Why do all rows of Pascal's triangle add to powers of 2?
- Primes in Pascal's Triangle
- Pascal's Triangle
- Numberphile
- Mathematical Secrets of Pascal’s Triangle
- Pascal's Triangle Song
- TED-Ed
- Art of Problem Solving
- Binomial Coefficients in Pascal's Triangle
- Binomial Expansion Using Pascal's Triangle
- Raise Binomials to High Powers
- Pascal's triangle for binomial expansion

Notable Thinkers

Mathematics

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