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The Law of Large Numbers in Gambling

Ilija Jaksic December 22, 2023
The Law of Large Numbers in Gambling

Have you ever wondered why some gamblers seem to have a knack for beating the odds? The secret may lie in understanding a crucial principle of probability theory—the law of large numbers, sometimes abbreviated as LLN. This law dictates that the actual outcomes increasingly align with expected probabilities as the number of trials increases.

The LLN is not just academic; it’s a practical tool for improving your gambling strategies. In this article, we’ll see how LLN shapes sports betting and gambling decisions.

What Is the Law of Large Numbers?

The law of large numbers was first conceptualized by the Swiss mathematician Jacob Bernoulli in the 17th century. It demystifies the behavior of outcomes in repeated, independent experiments, illustrating that as the number of trials increases, the average of the results approaches the expected value.

Consider the act of flipping a coin, which has two potential outcomes—heads or tails. Theoretically, each outcome has a 50% chance of occurring. If you flip the coin 10 times, the outcome might not be an exact 50-50 split. However, as you increase the number of flips to 1,000, the proportion of heads to tails will draw nearer to the anticipated 50% for each.

LLN exists in two forms—strong and weak. The strong law suggests that, given an infinite number of trials, the results’ average almost certainly converges to the expected value. In contrast, the weak law implies that for many trials, there’s a high probability that the average of the results will be close to the expected value.

In gambling, the law of large numbers plays a critical role. It explains why casinos always have an edge in the long run. For example, in a game with a slightly skewed probability in favor of the house, the more the game is played, the closer the average winnings of the casino will be to this expected edge, confirming the adage, “The house always wins.”

The Law of Large Numbers in Roulette and Slot Machines

In the classic game of roulette and slot machines, LLN plays a crucial role in determining the outcomes over an extended period. For instance, there are 37 pockets in European roulette, and the probability of the ball landing on red or black is 48.65%. Over a short series of spins, the results can deviate significantly from this percentage. However, as the number of spins increases, the actual outcomes will align more closely with the theoretical probability.

Similarly, in slot machines, each spin is an independent event governed by a random number generator (RNG). Despite common misconceptions, these machines do not get “hot” or “cold.” The longer the session, the more likely RTP will approximate the machine’s programmed payout rate.

How Does the Law of Large Numbers Work in Sports Betting?

In sports betting, employing the law of large numbers for analyzing vast data sets, such as past performance records and statistics, can aid in formulating more educated predictions about game or event outcomes. For instance, if a basketball team maintains a 60% win rate over a season, LLN might suggest a similar rate moving forward under comparable conditions. Yet, it’s essential to recognize that LLN doesn’t assure a definite outcome, as unpredictable events can still influence results. Bettors are advised to integrate LLN as a component of a broader array of analytical tools and betting strategies.

The intrigue with LLN in sports betting lies in its application to scenarios where statistical probability isn’t the sole influencing factor. Paradoxically, the more challenging it becomes to predict an outcome, the more tempting it is to lean on LLN. 

The law of large numbers in sports betting strategy is exemplified in the Martingale system, where the theory posits that the probability of winning increases with each consecutive loss. This concept has given rise to the “progressive draw strategy” in sports betting, particularly popular in national team competitions like the FIFA World Cup.

The strategy involves backing draws in consecutive matches and doubling the stake after each loss. It operates on the premise that it’s highly improbable for a tournament to conclude without a single draw. However, this approach can be fraught with risk. For instance, in the 2014 FIFA World Cup, the first draw occurred only in the 13th match. Had a bettor started with a 10-unit stake and followed this strategy, they would have had to place a staggering 40,960 units by the 13th game, accumulating a loss of 40,950 units before this point.

Common LLN Misconceptions in Gambling

Gambling myths and misconceptions often cloud players’ judgment, leading to irrational betting decisions. Gambler’s and hot hand fallacy are two of the most common fallacies rooted in these misconceptions. Both stem from a misinterpretation of the law of large numbers and the independence of each event in games of chance.

The Gambler’s Fallacy

One of the most striking examples of the gambler’s fallacy occurred in 1913 at a Monte Carlo casino. On this fateful day, the roulette wheel landed on black 26 times in a row. After the 15th black, players convinced that a red was due started betting heavily on it. As the black outcomes continued, more and more people bet on red, falling prey to the belief that past outcomes influence future events. This belief, however, defies the principle that each spin of the roulette wheel is an independent event. The phenomenon became known as the Monte Carlo fallacy, underscoring that in games of chance, each round is entirely separate from the last.

The Hot Hand Fallacy

Similar to the gambler’s fallacy, the hot hand fallacy is the belief that a streak (like consecutive scores in basketball or repeated appearances of a number in roulette) will continue. This fallacy overlooks the randomness inherent in each independent event, assuming that a short-term pattern will persist. In reality, each game or round offers a fresh start, with previous outcomes holding no sway over future ones.

Tips for Using the Law of Large Numbers in Gambling

Incorporating the law of large numbers (LLN) into gambling practices requires a blend of statistical understanding and disciplined decision-making. Here are some practical tips:

  • Understand sample size significance: In gambling, this means not expecting short-term play to reflect the statistical probabilities immediately. For example, slot machines with a 95% RTP might not pay out 95% in your first 100 spins, but it’s more likely to approach that percentage after 1,000 spins. 
  • Bankroll management: Allocate your funds. LLN suggests that outcomes will average over time, but this requires a bankroll that can withstand the variance in the short term.
  • Avoid pattern overreliance: While LLN helps in understanding long-term probabilities, avoid the fallacy that patterns in past outcomes can predict future events. Each spin, roll, or hand is independent in most casino games.
  • Probability-based decision-making: Make rational decisions and bets based on statistical probabilities rather than hunches or superstitions. For instance, in blackjack, use a basic strategy derived from the probabilities of different outcomes.

LLN as Your Ace in the Hole

The law of large numbers teaches us that outcomes will average over time, but each event is independent. Embracing this concept helps you make more informed decisions, manage your bankroll effectively, and avoid common fallacies and pattern overreliance. Let the knowledge of LLN be your guide, not your guarantee, as you continue to learn and practice responsible gambling.

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