What Is the Law of Large Numbers?

July 27, 2023
The law of large numbers is a theorem that says that the more times an experiment is performed, the closer it will come to its expected probability. This is because as more trials are conducted or observations are made, outliers and events which can bias the data will be minimized. The law of large numbers is part of probability theory, a branch of mathematics concerned with quantifying the potential outcome of an event. Although probabilities can not be used to accurately predict the future, they are helpful in modeling certain trends or behaviors. For this reason, the law of large numbers can then serve as evidence to support those conclusions without having to measure every occurrence within the population.

How Does the Law of Large Numbers Work?

In any experiment where there is a predictable outcome, it is not possible to guarantee the short-term output. This is because there may be fluctuations or outliers which can skew the average. However, as additional trials are performed, those fluctuations will make less of an impact and gradually filter themselves out. Eventually, the outcome will gravitate towards the mathematical probably.

The same can be said about the nature of a population. Again, it would be impractical to measure or observe every instance within the population. However, given a great enough representative sample size, we can say that our predictions about the population are reasonable due to the law of large numbers.

Law of Large Numbers Example

Suppose you wanted to test the probability of a 50/50 coin flip. A coin flip can only have two outcomes: heads (H) or tails (T). Since coins are evenly distributed in weight, it’s reasonable to assume the probability of either outcome is 50 percent.

In theory, you know that you should be able to get heads about every other trial. However, anyone who has ever flipped a coin knows that this won’t occur perfectly half the time. For example, you might conduct ten trials and get the following results:

T, H, H, H, T, H, T, H, H, H

In this particular situation, you got heads for seven out of the ten flips resulting in a ratio of 0.7. However, we know that the theoretical probability should be 0.5. Therefore, we continue with additional trials.

Now suppose in the second iteration of ten trials, the coin flips resulted in the following:

H, T, T, H, T, H, T, H, T, T

In this second series, heads appeared less often than tails resulting in a score of only 0.4. However, if we combine all 20 of these measurements together, then we get 11/20 or 0.55. Hence, the outcome is now getting closer to the expected probability of 0.5.

This same experiment could be carried out indefinitely with many more trials. Sometimes there might be more heads than tails or vice versa. Yet, as these measurements get combined with the existing pool of data, it will gradually produce the expected outcome of 50 percent.

Hence, this is the law of large numbers at work. Given enough trials and observations, the bias of outliers and short-term streaks will eventually be minimized as the total population of data starts to become more representative of the mathematical expectation.

The Gambler's Fallacy vs the Law of Large Numbers

The law of large numbers should not be confused with the gambler’s fallacy. This is a false conclusion drawn from short-term trials that an event is likely to occur. However, this determination does not align with the probable outcome.

For example, suppose in this previous example with the coin toss, you were asked to place a wager on what the next outcome would be. Because there was a streak of more heads than tails, then you might be inclined to bet “heads” because that’s the outcome that has occurred more often.

Yet, this belief would be unfounded. In reality, each new trial has the same 50/50 probability despite what happened in the trials before it. Therefore, it's misleading to think that just because heads came up more often that the same pattern will continue.

This is why roulette tables at casinos will often have a display next to them showing the last dozen or so numbers that were spun. They are in effect using the gambler’s fallacy against their guests by supplying them with the data to draw these false conclusions.

For instance, just like the heads or tails coin flip, a roulette player might make the same mistake in picking red or black. They may see that black has come up more times than red and come to some determination about what the next spin should be. However, the roulette wheel always has the same probability with every spin. It does not “remember” how many times red or black appeared in the past. Therefore, in the short run, there is no telling what the next outcome will be.

The law of large numbers explains why when it comes to casinos, the house always wins. In the case of roulette, there are also two green squares in addition to the 36 red or black squares. Therefore, when a person bets on red or black, their odds are not 50/50 but really 16/38 while the casino enjoys odds of 18/38. Therefore, given the law of large numbers, the longer someone plays, the more likely the casino will eventually win.

In fact, this same explanation can be expanded to every game in the casino. Because they have thousands of guests every day with limited amounts of money, it's highly probable that more people are losing than winning. Hence, the law of large numbers explains why the gambling industry is so lucrative.

The Insurance Industry

Another industry that benefits from the law of large numbers is insurance. Insurance contracts are sold to customers as a way of financially protecting their vehicles, homes, lives, etc. Yet, the probability of these customers ever having to make a claim is relatively low.

Of course, accidents do happen and insurance companies will make payouts when it's required. However, the reality is that this happens a small number of times relative to the vastly large pool of customers who pay premiums but never make a claim. Hence, because the odds are in the insurance company’s favor, the law of large numbers can be leveraged to their advantage.

Law of Large Numbers and Business

In business, the law of large numbers is sometimes used casually with an alternate implication that a business cannot grow indefinitely forever.

An example of this would be the early years of the digital streaming service Netflix. In the 2000s, Netflix was attracting new subscribers at a record-breaking pace year over year. The company was enjoying a massive influx of revenue as it grew in popularity, and investors were very excited about the prospects of how valuable this company could become.

However, skeptics cited that due to the law of large numbers, this growth was not sustainable. And over time, they were correct. This is because intuitively, there are only a certain number of people on the planet, so there is a maximum of how many subscribers there could ever possibly be. Additionally, it was only a matter of time before other streaming competitors entered the field and began to steal market share. Therefore, it was inevitable that the company would eventually stop gaining new subscribers at such an accelerated pace.

It should be noted that this use of the law numbers does not carry the same meaning as the statistical definition.

The Bottom Line

The law of large numbers says that if an experiment is repeated enough times that it will eventually achieve its probable outcome. This can be used to draw conclusions about the population as a whole or model certain trends.

However, it's important not to base your assumptions on too small of a data set. This would be the gambler's fallacy and cause you to make predictions that are irrational. When an outcome has the probability with each new trial, we know from the law of large numbers that bias will eventually be minimized, and the outcome will arrive at its mathematical probability.