The Law of Large Numbers Explained with Examples

By Jerry Smith Staff Writer Updated: 03/30/2023

Jerry Smith Staff Writer See Full Bio

Jerry’s greatest advantage is his extensive experience on the casino floor. His time as a casino manager taught him everything about what makes a player tick. Besides being a skilled poker player, he has deep knowledge of all live table games and gambling regulations in the US.

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The Law of Large Numbers

• The Law of Large Numbers: An Introduction
• The Law of Large Numbers Defined
• Weak Large Numbers
• Strong Large Numbers
• The IQ Simulation
• The Coin Flipping Example
• Frequently Asked Questions

The law of large numbers explained: that’s what today’s blog post is going to cover. While the law of large numbers may seem a complex topic at first, it’s important to realize that once you start breaking things down, it’s actually a lot simpler than you probably think.

The Law of Large Numbers Explained

The law of large numbers is a famous concept within probability theory. It states that as the number of trials or observations increases, the average number of the results will approach the expected value. This means that, in general, the larger the sample size, the more accurate the prediction or estimate will be.

In terms of gambling, the law of large numbers relates to the house edge; the amount of money that a land-based or real money casino sites can expect to make from a game in the long run. In this post about the law of large numbers explained, we’re going to show you all you need to know about the theory.

The Law of Large Numbers Defined

The law of large numbers defined: “The law of large numbers is a theorem in probability theory that states that as the number of independent, identically distributed trials or observations increases, the sample mean of the observations will converge to the expected value of the population.”

The rule of large numbers plays a significant part in gambling, as you’ll see from this page. If you’d like to learn how to claim the best bonuses before you start gambling online, make sure to check out our dedicated online casino bonuses page.

The Weak Law of Large Numbers

When it comes to the law of large numbers, there are actually two forms. However, these differences are mainly theoretical. The first type is the weak law of large numbers, and this theory states that as N increases, the sample statistics of the sequence converges in probability to the population value. The weak law of large numbers is also sometimes referred to as Khinchin’s law.

So, what does the weak law of large numbers really mean in practical terms? The weak law of large numbers is a statistical principle that says that as the number of observations in a sample increases, the average of those observations will become closer and closer to the expected value.

This means that if you flip a coin a large number of times, the proportion of heads you get will eventually approach 50%, which is the expected value. The weak law of large numbers is a fundamental concept in statistics and provides the basis for many other statistical theories.

The weak law of large numbers theory applies regardless of sample size; however, as you reach infinity, the probability of the law being met exactly increases exponentially.

The Strong Law of Large Numbers

The strong law of large numbers is a more powerful version of the weak law, stating that the sample mean will converge almost surely to the expected value, rather than just in probability. In other words, as the number of trials or observations approaches infinity, the sample mean will converge to the expected value with probability one, meaning that the convergence is certain to happen.

When it comes to gambling, the strong law of large numbers means that, over time, as a player places more bets, the likelihood of the house edge being met is almost certain.

Professional gamblers are able to use the strong law to make mathematical decisions, particularly when it comes to placing informed bets and predictions. When dealing with the strong law of large numbers, the mean will also converge on the population mean once the sample size begins to increase. To some, the strong law of large numbers is known as Kolmogorov’s strong law.

Law of Large Number Simulations: IQ

There are several laws of large numbers example simulations that can be used, although the two most common are the IQ and coin flipping examples. The IQ example is relatively easy to understand. Picture that a study is analyzing IQ scores, and 100 participants are randomly selected to take part in the trial. The goal of the trial is simply to measure their IQs.

Firstly, the actual IQ of each 100 participants is measured. Once this data has been collected, a recalculation is done, with the sample mean versus every additional person that’s taking part in the trial. Once complete, this process will have produced a sequence of sample mean that grows in size from one to 100.

If the law of large numbers is to be believed, the sample mean would grow on the population mean as the number of new participants is added into the mix. At the beginning of the study, the results are all over the place. However, as more participants give their score, everything equals out, showing that the law of large numbers is indeed an accurate theory.

Coin Flipping

The coin flipping example is another popular way of showing the law of large numbers in practice. Suppose we flip a fair coin 100 times and record the number of times it lands heads up. According to the law of large numbers, as we increase the number of coin flips, the proportion of heads observed will get closer and closer to 0.5, which is the expected value of a fair coin.

For example, if we repeat the experiment 1,000 times and record the proportion of heads in each trial, we would expect the proportion to be closer to 0.5 on average than if we only repeated the experiment 10 times. And if we repeated the experiment an infinite number of times, the proportion of heads observed would converge to 0.5 exactly.

This example highlights the power of the law of large numbers in helping us understand the behavior of random events and the ways in which sample sizes can impact the accuracy of our predictions. This is where the law of large numbers comes into play with casino games. Check out the table below to see the expected house edge for different casino games, such as the house edge for online blackjack.