Probability Machine (Francis)2

A Simulation of the Random Nature of Stock Prices

Probability Machine (Francis)2

Many measurements seem random, such as the heights of humans, the length of leaves, the roll of 5 dice, and the change of stock market prices. But when academics, statisticians and mathematicians view the world, they see patterns that others do not notice.

The device like the one shown in the video below was first created by Francis Galton in about 1890 and today has many names; Probability Pinball, Galton Board, Quincunx Board, and Bean Machine. The Galton Board simulates random events and how the distribution of a large number of events tend to form a bell shaped curve. The beads falling through the pins are not predictable as to which course they will take, but the average outcome of a large number of dropping beads, just like 780 monthly returns of Index Portfolio 60, form a relatively predictable pattern. In 1794, Johann Carl Friedrich Gauss described the pattern as a normal distribution, referring to the fact that it is normal for random events to form such distributions.

Gains and Losses are Impossible to Identify in Advance

Watch this video from the Khan Academy on the concept of the Normal Distribution.

The display of a central distribution around the average (Central Limit Theorem) is indicative of the randomness of the news that generates the random and unpredictable movements of the S&P 500 or any other index. Watch this video from the Khan Academy on the concept of Central Limit Theorem.

See the similarity of the distributions above to the 5 dice roll distribution on the right side of Figure 4-16 below. The characteristics of the average and standard deviation are not the same as the 964 monthly returns of the S&P 500 index, but the concept of a central distribution around the mean is very similar, as you see when they are compared. Also note the sampling error in the 5 year (60 month) periods as compared to the 964 months, which are very similar to the sampling error of only 60 roles of the dice, versus 1,000 rolls. The randomness of the news generates the random and unpredictable movements of the S&P 500. In a random distribution, successful market timing over the long term is impossible.


Based on the following histograms, investors can see how difficult it is to find the randomly distributed days with gains or to avoid the days with losses. For each period, the large gains or losses for the entire period are highly concentrated at the right and left tails, making them impossible to consistently identify them in advance. In other words, it is impossible for time pickers to consistently outperform the market.






"A chance event is uninfluenced by the events which have gone before. If a true die has not shown 6 for 30 throws, the probability of a 6 is still 1/6 on the 31st throw. One wonders if this simple idea offends some human instinct, because it is not difficult to find gambling experts who... advocate ... the principle of 'stepping in when a corrective is due'." This is no different than stock market forecasters looking for the next "correction." Watch this video from the Khan Academy on the concept of the Law of Large Numbers vs. Averages.

Although we may have exposed you to more math and statistics than you cared to see, we hope that the message of the unpredictability of markets has sunk in. Otherwise, it is a very expensive lesson to learn the hard way.