What Is the Law of Large Numbers and How Does It Apply to Betting?

A Swiss mathematician, Jakob Bernoulli (1655 – 1705), established the law of large numbers. He discovered that the larger the sample size of the event, the more likely it is that you’ll know its true probability. However, many bettors still fail to understand the concept, which is why so many people fall for the Gambler’s Fallacy.

Fortunately, once you grasp the law of large numbers, you can stop making common betting mistakes. It is a mathematical law that increases your knowledge of the laws of probability and enables you to stand firm psychologically when your betting system goes through a rough patch.

This article outlines what the law of large numbers says, how it applies to gambling and how it helps you learn realistic expectations when betting.

What Is the Law of Large Numbers?

The law of large numbers is a way of determining a random event’s probable and expected outcome. It investigates the result when you conduct the same experiment a huge number of times. Ultimately, the more often you repeat an action, the closer the results are to the expected value.

Bernoulli used a coin toss as an example. Assuming you use an unbiased coin, the chance of it landing on heads is 50%, and the chance of it landing on tails is also 50%. However, if you toss a coin 50 times, it probably won’t have an even 25-25 heads and tails distribution. Indeed, the number of heads could be significantly more than tails, say 40-10.

You can expect either heads or tails to land at least six times in a row in any sample of 50. Once the sample reaches 1,000, the coin may land on heads, for example, 10+ times in a row. This fact can ensure the coin comes up heads significantly more often than tails in a small sample.  

Over time, though, once the sample size gets large enough, such as one million, there should be a near-equal distribution of heads and tails. At this point, the expected deviation will be up to 500, which may seem like a lot but is very small in percentage terms.

Suppose the coin lands on heads 500,200 times and tails 499,800 times. This means heads appeared 400 times more than tails. However, this equates to 50.02% heads and 49.98%, a tiny differential.  

How to Apply the Law of Large Numbers in Sports Betting

Once you understand the law of large numbers, it can help you avoid falling into the Gambler’s Fallacy trap. You now realize that there are sometimes surprisingly large deviations from expected results in the short term.

The infamous incident in Monte Carlo in 1913, where the ball landed on black 26 times in a row on the roulette wheel, is an example of extreme deviation from expected results. In one of the clearest examples of Gambler’s Fallacy on record, bettors continued to pile money on red, believing they were more likely to win on the next spin. Of course, the outcome is random, so each spin is independent of the one before and after.

Therefore, the sample size is key according to the law of large numbers. You should not expect an ‘even’ spread within a small sample size. The roulette players in Monte Carlo on that fateful day discovered, to their great cost, that 26 spins of the wheel were nowhere near enough to bring parity between red and black.

As far as the law of large numbers in sports betting is concerned, you should apply the principle to any systems you have in place. Suppose your system’s average win rate is 50%. During a short period of, say, 100 bets, you could, and possibly will, experience significant variance in results.

It is unlikely that you will win precisely 50% of the time. You must also account for your betting system’s sample size. For instance, if the win rate is calculated from 250 bets, it is likely not representative of how often it picks winners. On the other hand, if your system’s sample size is 2,500, a 50%-win rate is probably quite close to reality.

In a nutshell, don’t place a massive amount of faith in a betting system, or tipster, with a sample size of a couple of hundred bets.

What About the Law of Large Numbers in Casino Betting?

The law of large numbers is arguably even more relevant in casino games than sports betting. After all, when betting on sports, it is difficult to determine the true probability of an outcome. Casino games are different because you know the statistical likelihood of an event occurring.

For example, the chances of the ball landing on red in European roulette is 48.65% each time. However, as the Monte Carlo example showed, you should not ‘expect’ the ball to land on red 48% or 49% of the time in the next 100 spins. After 1,000,000 spins, the ball should land on red close to 48.65% of the time.

Unfortunately, this information is of little help to you in accurately predicting the next spin. On the plus side, it does prevent you from making the classic mistake of thinking the ball is ‘due’ to land on a certain color or number.

The law of large numbers is also an important consideration for slot gamers. Suppose the RTP of a specific game was just 74% in the last week when the overall average is 96.5%. Does this mean the game is ‘due’ to pay out at a higher rate this week? NO! Eventually, it will have a period of higher RTP to bring up the average, but this scenario may not occur for hundreds of thousands of spins.

So, if you’re in a brick-and-mortar casino and have lost a lot of money on a slot machine, don’t get angry because someone started playing when you left to get change! It is illogical and mathematically inaccurate to believe a win is more likely to happen for them than you. Sure, they might win, but it is a matter of luck, as it always is with games of chance.

Don’t Expect Profit from A Positive Expected Value

The biggest thing you should take from the law of large numbers in sports betting is the necessity of patience. Even if you consistently find value bets, it doesn’t mean you’ll win in the short to medium term. The false belief that achieving a positive expected value (EV) on a wager means earning a profit has disappointed many bettors.

We can go back to the coin toss to elaborate on why you must prepare for losses even when you have the edge over the bookie. The chance of an unbiased coin landing on tails is 50% each time, the equivalent of 2.00 in decimal odds. Let’s say you get odds of 2.10 on tails appearing each time. This wager has the following mathematical edge:

(0.5 x 1.1) – (0.5 x 1)

0.55 – 0.5 = 0.05

You can expect to win 0.05 units for every unit you risk. This equates to an edge of 5%. In betting parlance, this sizable edge will ultimately result in long-term profit. However, you could experience disappointing losses in the short to medium term.

The Law of Large Numbers in Action

Imagine setting your bankroll in a way where one unit is $100. While the expected value of each coin toss is $105, you won’t win that amount after the first toss. You will either be up $110 or down $100. Ultimately, there’s a 50% chance of you losing money after a single coin toss, a risky investment for sure.

The number of possible outcomes increases to three after two tosses of the coin. You can be:

1. Up $220 (The coin landed on tails both times)
2. Down $200 (The coin landed on heads both times)
3. Up $10 (The coin landed on tails once and heads once)

There is a 25% chance of scenario one occurring, a 25% chance of scenario two occurring, and a 50% chance of scenario three happening. At this early stage, the probability of being in a losing position is now 25%, an improvement on the situation after toss #1 but still far too high for any sensible investor.

Less and Less Risky

Fortunately, the risk dwindles the more often you toss the coin. After 1,000 coin tosses, the expected value is $5,000 ($5 x 1,000), but you’re as likely to be below that mark as above it. On the plus side, the chances of being in a losing position at this point are approximately 7%.

Once you get to the 5,000 coin toss mark, there is only a 0.04% chance (decimal odds of 2501.00) of being in a losing position. Moreover, there is a 50% chance of your profit being at least $25,000, which is the expected value ($5 x 5,000). If you’re somehow in a losing position at this point, you are extraordinarily unlucky.

This example shows the benefit of waiting for a large sample size before committing to any specific betting system. After one coin toss, there’s a 50% chance of losing money. After 1,000 coin tosses, this risk drops to 7% and is a miniscule 0.04% after 5,000 tosses.

Incidentally, the reverse is true with bets that have a negative expected value, like red or black, on a roulette wheel. Bettors can, and have, won substantial sums of money in the short term. Perhaps some of our readers have experienced that legendary night when they won $10,000 on roulette!

However, the more you play, the more likely you’ll give the money back to the casino. If you have a great run on the roulette wheel, it is probably best to pocket the money, leave, and steer clear in the future!

You’re Not ‘Fated’ To Win or Lose

If you understand the law of large numbers, you know that there are no ‘guarantees’ of winning OR losing, no matter the odds. You realize that huge degrees of variance are possible in small sample sizes. However, the bigger the sample, the smaller the deviation from the expected value. Eventually, you will get quite close to the true probability of an outcome.

It is also important to avoid common betting mistakes. Traps we sometimes fall into include false syllogisms and confirmation bias. The former is a flawed form of reasoning.

Here’s an example: Michael is an engineer, and engineers enjoy partying, so Michael enjoys partying. An example of a false syllogism in gambling is to suggest that you have won your last five wagers. The last time this happened, you lost, so you will lose again this time. Those who understand the law of large numbers know that each event is unrelated to the last one.

Confirmation bias, on the other hand, involves looking for information that fits an opinion you already hold. Suppose it is game 7 of the NBA finals between the Lakers and the Celtics. You search for game 7 trends to see if one team has an advantage. In reality, there is nothing concrete to favor either team. However, because you’re a Lakers fan, you focus on the trends that point to a Los Angeles victory and ignore anything that suggests a Celtics win.

In the end, learning about the law of large numbers in sports betting can make you a more effective bettor. Through this process, you get a firm grasp on mathematical probability and are better prepared to handle poor betting spells, which are inevitable.