Law of Large Numbers and the Gamblers Fallacy[Math Mondays]

A simple idea that gets misunderstood a lot.

Sep 11, 2023

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Time to cover something that everyone and their grandma has come across,

And pretty much everyone gets wrong.

We covered how online Casinos make a lot of money in this post. Misunderstanding this idea is one of the reasons that happens. In this edition of Tech Made Simple, we will be covering the Law of Large Numbers, how it gets misinterpreted, and what it actually says.

Law of large numbers - Wikipedia — This simple image is the root of one of the common fallacies in the world. Image Source

According to economists Hersh Shefrin and Meir Statman, investors tend to hold onto stocks that have depreciated and sell stocks that have appreciated. They call this a “general disposition to sell winners too early and hold losers too long.”¹⁰
Investors may see the continual rise of a stock’s value as an indication that it will soon crash, therefore deciding to sell. Likewise, if a stock has lost value, this can be taken as an indication that it is due to appreciate, and so they decide to hold onto those stocks. Gambler’s fallacy may be at work here, as investors are making decisions based on the probability of a fairly random event (the stock’s price) based on the history of similar past events (the trend in its previous price points).
-A non-gambling related example of the Gambler’s Fallacy, to show you show universal it is. Source

An Intro to the Law of Large Numbers

What is the Law of Large Numbers- The law of large numbers is a theorem in probability theory that states that as the number of trials in an experiment increase, the average of the results will get closer and closer to the expected value. In other words, the law of large numbers says that if you repeat an experiment many times, the results will tend to average out.
Why it’s amazing- The law is very important in the scientific fields. It allows us to make predictions about complex phenomenon using relatively simple experiments. Entire industries (gambling, insurance, etc) have sprung from the fact that we can use this law to make long-term predictions on very chaotic events.
What People Get Wrong- The most common misconception here is that the law of large numbers guarantees that you will eventually get the expected value. This is not true. The law of large numbers only says that the average of the results will converge towards the expected value as the number of trials increases. It does not guarantee that you will ever get the expected value, especially for any finite number of trials (even if that finite number of trials is extremely high). This is what leads to the Gambler’s Fallacy. The law of large numbers only makes predictions on the convergence of infinitely many experiments- it can’t be used to make predictions about a specific experiment.

The law of large numbers is one of those areas where mathematical precision is extremely important to understand the nuances of the law. If you want more details on it, the below video is a great introduction to the core ideas, without getting too mathy.

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