The Law of Large Numbers in Gambling

Despite the fact that success in online sports betting is not only attributed to luck and pure statistical probability, it is understandable that similar to everything else in the universe, betting is also governed by the laws of physics.

Before you decide to shun this statement as pure hogwash, we need to travel back to 17th century Switzerland and see if one of the most controversial theorems in gambling, the law of large numbers, is indeed applicable to sports betting.

What Does the Law of Large Numbers Mean?

Don’t worry. There's no reason to dwell upon physics or math here, nor do you have to be a science expert to understand everything. On the contrary, the Law of Large Numbers (often abbreviated as LLN) is rather simple and can be easily tested.

The LLN is part of the probability theory and examines the result of carrying out the same experiment over a large number of repetitions. The more times you repeat an action, the closer its results should appear to the expected value. Imagine tossing a coin. There are two possible outcomes, heads or tails and each outcome has a theoretical 50% confirming percentage.

Now, if you toss the same coin 10 times and write down the results, it's highly probable that the 50% percentage will not apply. In a 10-time experiment the result might even be 10 consecutive heads with no tails at all. However, if you continue tossing to say, 100 times, the results will start to even out. If you toss 1,000 times, you’ll get an even more balanced number.

The more you flip the coin, the closer the results will get to the 50% average. The coin toss is the simplest experiment for confirming the law of large numbers and is often called a Bernoulli trial, named after Jacob Bernoulli, a 17th century AD Swiss mathematician and mechanic.

The formalization of the Bernoulli trial is called the Bernoulli process. Bernoulli came from a mathematician family and is famous for his saying “in the end, there’s equilibrium in the universe”.

The Gambler’s Fallacy

So lets put Bernoulli's theory into practice. Given that a repetition of a certain two-aspect action will offer 50% results in the long term, this surely means that after 10 consecutive head tosses, the 11^th one should land as tails, right? Far from it. We tend to rely on the Law of Large Numbers to predict gambling outcomes, due to our need to explain the world through repetitive patterns.

This theory has been followed countless times on casinos, especially on the “red-black” roulette bet, which is also split between two outcomes. Gamblers tend to apply Bernoulli's trial in roulette believing that, if an aspect has happened way too many times in a row, it is bound to offer the opposite results in the next spin.

This is described as a fallacy of the maturity of chances and although theoretically correct according to the LLN, is not confirmed in practice. You’ve probably bumped into this theory, described as “gambler’s fallacy”. There’s also a more specialized name, the Monte Carlo fallacy.

It is named after an incredible story that took place at the glamorous Monte Carlo casino on August 18^th, 1913. At some point in the night the roulette ball had landed on black a few times in a row. After the 8^th time, some perceptive gamblers got interested and started pushing money on red, as according to gambler’s fallacy, black had significantly exceeded its confirming possibilities.

As the spins showed black for a few more turns, nearly every man and woman in the casino had their eyes fixed on the roulette wheel, while even more gamblers were now maniacally backing red. No one had shared his thoughts on where to place their money.

They all acted according to instinct; an instinct that was based on the Gambler's Fallacy. Believe it or not, the ball landed on black for 26 consecutive spins, against any probabilities or odds. The entire amount lost during that night was never publicly admitted, but rumour has it it surpassed one million French francs, the official Monaco currency at the time.

This was a total debunking of the theory and a permanent reminder that an ounce of practice is worth a ton of theory. Also it shows that you can not predict future results based entirely on past outcomes.

Bet With the Highest Odds

Does the Law of Large Numbers Apply in Sports Betting?

The reason we analyze the Law of Large Numbers (LLN) is to examine if it can be applied to sports betting, where various parameters besides statistical probability can influence an event. What is odd, is that the lesser our chances of predicting an outcome are, the more we are tempted to follow the LLN.

In fact, the (in)famous Martingale in sports betting is based entirely on this Law, as in theory the probability of winning your next bet is augmented with every consecutive loss. How can this be applied to sports betting? A popular betting system has been conceived called the progressive draw strategy.

All you have to do is back draws in consecutive matches and double your stake each time you lose. This is popular especially in national team’s competitions, like European Championships or a FIFA World Cup, when almost all the matches (barring the last group ties) are scheduled on different times.

Betting on draws means that you put your money on minimum 3.00 odds, that can reach even higher in matches with strong favorites. You don’t have to analyze news, recent form or other stats, just put an initial stake on the draw and wait for it to be confirmed. If the first match does not end up as a draw, you should place another draw bet on the next one and double your stake.

Let's say you start with a 10-unit stake and back draws priced at 3.00 odds.

1. 10 units on a draw bet on the first match | bet lost | profit -10 units
2. 20 units on a draw bet on the second match | bet lost | profit -30 units
3. 40 units on a draw bet on the third match | bet lost | profit -70 units
4. 80 units on a draw bet on the first match | bet won | profit 90 units

It’s highly unlikely to witness an entire tournament end without a single draw, but you also can't know when the first one will appear. For example, in the 2014 FIFA World Cup, the first draw was recorded in the 13^th match that took place between Iran and Nigeria.

If anyone were following the progressive draw strategy and had started with a 10-unit bet, he would have to place 40,960 units on the draw while knowing he had already lost a total of 40,950 units. There are two questions here.

Do you have a large enough bankroll to keep following this strategy after 10 lost bets? And even if you do so, are you truly willing to risk losing even more? Following the Kelly Criterion in sports betting, you can determine the ideal stake based on your bankroll. 

How Do Gambling Sites Apply the Law of Large Numbers

The sole winner of the Law of Large Numbers is the casinos and bookmakers. The house always wins, saying is based on this theory. After millions of roulette spins, card shuffles or football matches, the initial odds will be confirmed and along with them the overround will snatch your bankroll.

Imagine you are tossing a coin and each time it lands as heads, you get paid with 1.90 odds. In the long run, you'll have a 50% winning percentage and a 10% loss to your bankroll. If you want to know how to beat the bookies, you should search for valued odds that exceed the confirmation probability.

In the example above, if you could agree into getting a 2.10 odds for every coin toss win, you would be profitable in the long term. Seeing that no bookie or casino will offer you these odds, it's up to you to do the research and find some profitable mistakes.