Pascal’s Triangle: order out of chaos

Posted by Robin Powell on July 9, 2018

“Essentially, all models are wrong,” the famous statistician George EP Box once said, “but some are useful.” A good example is the Galton Board, designed by the Victorian genius Sir Francis Galton in the 1870s to illustrate the concept of the normal distribution.

The design consists of a vertical array of pegs, ordered according to a quincunx pattern. A large number of beads, introduced from a centre point at the top of the board, make their way through the pattern, eventually coming to rest in one of several bins uniformly arranged across the bottom.

Every time one of the beads hits a peg, it goes left or right with equal probability. So, for a bead to end up in one of the outer bins, it needs go the same way every time. In the middle, however, there are many different paths a bead can take. As a result, the beads tend to head towards the centre of this distribution.

Over the last few weeks we’ve been featuring a series of three videos on the Galton Board and the lessons we can learn from it. The third and final video, which specifically looks at Pascal’s Triangle, is now online. Named after the French mathematician Blaise Pascal. This triangular array of the binomial coefficients further helps to explain the normal distribution’s bell-shaped curve.

These videos have met with wide variety of responses from the investment community. Although they’ve been generally well received, some commentators have pointed out that investment returns don’t conform to the normal distribution.

Strictly speaking, that’s true; investment returns don’t form a perfect bell curve. Without wishing to get too technical, there are thin-tailed and fat-tailed distributions. But, as a model for long term investment returns, the Galton Board is very useful.

Why? Because markets are broadly efficient, and your chances of beating them consistently are very slim.

The stock market aggregates the knowledge, skill and expertise of millions of traders and investors around the world. Through the trades they make — and there are millions of them every second — they agree a fair price for every stock.

The price goes up and down randomly, in response to new information, which by its nature is unknowable. Once you’ve built in a risk-appropriate daily return of around 0.03% a day (based on equity returns over the last 50 years), there’s a 50:50 chance that the price of a particular stock, or the market as a whole, will rise or fall each day.

In other words, for every fair price:

— the most likely outcome is a fair return;

— there is an equal chance that future returns will be below or above a fair return;

— the further the monthly return is from a fair return, the less likely it is to occur:

— an extremely negative return (a black swan) is as likely to occur as and extremely positive one (a green swan).

In conclusion, then, although returns are random, there is a pattern to them. Invest for long enough, and you can expect to receive a fair return.

In 1889, Sir Francis Galton wrote this: “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. It reigns with serenity… 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.”

Here’s the full set of videos. Please do share them if you find them helpful. Finally, I’d like to thank Mark Hebner at Index Fund Advisors for his help with this subject. You’ll find out plenty more about it on his website or by visiting

Galton Board: Normal distribution (Video 1 of 3):

Galton Board: Reversion to the mean (Video 2 of 3):

Galton Board: Pascal’s Triangle (Video 3 of 3):

The Evidence-Based Investor is produced by Regis Media, a boutique provider of content and social media management to financial advice firms around the world. For more information, visit our website and YouTube channel, or email Sam Willet or Christina Waider.

Robin Powell

Robin is a journalist and campaigner for positive change in global investing. He runs Regis Media, a niche provider of content marketing for financial advice firms with an evidence-based investment philosophy. He also works as a consultant to other disruptive firms in the investing sector.


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