Law of Large Numbers – The Gambler's Foe or Ally?

Law of Large Numbers and Probability in Gambling

The law of large number applies to series of randomly generated independent events with even probability of the possible outcomes. Coin tosses, roulette spins, and dice rolls are just a few examples. Practically, any event with unknown outcome classifies as a random event. For the purpose of this article, however, we focus on events that have a finite number of equally probable outcomes, where the outcome of each trial is not affected by the previous ones. The law of large numbers states that as the number of trials increases, their average result gets closer to their expected value.

Let’s illustrate this with an example from online gambling. If you are playing roulette, each spin is independent from the previous ones. The roulette wheel has 18 red pockets out of 37 (European and French). The number of blacks is the same. That said, the probability for both red and black is 48.65%. According to the law of large numbers, the more spins are completed, the closer the results of red and black will be to their theoretical probability. For straight up bets, the probability of every number, including the zero, is 2.70%. Looking at each pocket separately, the expected average over a very large sample will be very close to this value.

Weak Law of Large Numbers, and Its Strong Counterpart

Without getting too deep into math theory, we’ll mention that apart from the uniform law, there are two other theoretical forms. These are called weak and strong law of large numbers. They differ mainly in the definition of convergence of random variables. The weak law of large numbers uses convergence in probability, and the strong law of large number uses almost sure convergence. While these forms are important in advanced statistics, in the context of gambling, it is enough to remember the uniform definition.

Practical Implications of the Law of Large Numbers

Knowing the odds is not all in gambling. There are several conclusions from the law of large numbers that players must keep in mind at all times. Here’s what you should remember in order to avoid confusion and disastrous mishaps when playing games of chance:

• Size Matters – the law applies to very large samples, hence its name. You can’t expect it to be valid for a sample of 20 roulette spins.
• Deviations Are Possible – short term streaks of one type of outcome are possible and natural.
• Probability Is Not Obligation – even probability doesn’t mean even distribution of results over time. Again, convergence in probability is observed after a very large number of trials.
• Remember Independence – the probability of every trial is independent from past results.

As you can guess, many of the gambling mistakes and fallacies are related to failure to follow these guidelines. Strangely enough, doing the math doesn’t always help. Even players with good calculation skills tend to misinterpret probability and convergence, because they neglect the importance of the points above.

Common Gambling Mistakes

Gamblers often fall in traps due to misunderstanding of the law of large numbers and probability in general. People’s reasoning is plagued by all sorts of biases and heuristics that result in logical fallacies. To no surprise, the most notable one is called the Gambler’s fallacy. Check the paragraphs below to learn about the most common gambling mistakes that result directly from ignoring the law of large numbers. For a broader list of gambling mistakes that are related to other reasons, you can check our article dedicated to the mistakes that you can make when gambling.

Misunderstanding Probability

Many players believe that convergence to probability excludes streaks of the same outcome, e.g. roulette hitting red 10 times in a row. When they see a streak, they begin to suspect foul play by the house, because the result seems non-random. In fact, streaks are possible in small samples, and ten spins is a very small sample. Furthermore, casinos have enough advantage that’s built in the game rules to ensure long term profitability for the house, so there’s no need to rig the games.

Also, many believe that a streak must begin to even out, e.g., if the roulette ball hits red so many times, then black must be due. This is an underestimation of the independence of each trial. In fact, the probability for black in every next spin remains the same, which is a little less than 50%. Patterns observed in the past won’t make it a 100%. Neglecting that fact is, most of the times, the root cause of the gambler’s fallacy.

The Hot Hand Fallacy

Another common mistake related to streaks is the expectation that a given outcome is hot and will continue to appear. This is the so-called “hot hand fallacy”, which originally says that if a player is in a scoring streak, he’ll keep on scoring. This is also applied to roulette (hot and cold numbers), slots (hot and cold symbols), and any other game with various outcome combinations. In reality, hot numbers don’t have to keep coming up. Streaks can end at any given time, and then resume after couple of alternative outcomes. Randomness and unevenness go hand in hand. Just like the gambler’s fallacy, the hot hand fallacy stems from the underestimation of the independence of future outcomes.

Misunderstanding Joint Probability

This has a lot to do with the time when we calculate the odds. Let’s take roulette as an example again. The probability for black in a single spin is 0.4864 (48.64%). Now let’s say we are considering a series of upcoming spins and want to calculate the combined probability for black in the next three rounds. The result will be 0.4864³, or 0.1150 (11.5%). However, this doesn’t change the probability for any individual spin, which remains 48.64%. Also, this calculation doesn’t work in retrospect. If the outcome was black in the past two spins, the probability for black in the next one will be 48.64%, and not 11.5%. Therefore, you should remember the old saying that the roulette wheel has no memory. Forget past results think of every next spin as it is the first one.

Applying the Representativeness Heuristic

Heuristics are described as simplified rules or mental shortcuts used in decision making. They help in speeding up reasoning, especially when the collection and processing of the necessary info are too cumbersome. However, heuristics don’t produce optimal results, and lead to reasoning errors. The representativeness heuristic is very likely to be used in conditions of uncertainty. In a nutshell, when people see series of outcomes with no logical connection, they find this to be representative of randomness. If a pattern appears, even in small samples, people begin to think that the outcomes are not truly random. This insensitivity to the sample size forces gamblers to neglect the law of large numbers and trust the law of small numbers instead. Despite its name, the latter is not a law, but a logical fallacy, also known as hasty generalization.

Confirmation Bias in Gambling

In a nutshell, this is the human tendency to look for evidence that confirms our theories. Also, we tend to ignore disproving evidence. This is a way to perpetuate erroneous beliefs and can be extremely harmful. For instance, we may believe in the benefits of a betting system which has no statistical foundation whatsoever, regardless of series of losses.

Reliance on Factors Beyond our Control

Failure to understand and properly apply the law of large numbers, makes us susceptible to all sorts of beliefs and superstitions. For instance, players can put enormous faith in a betting system that’s based on expectations with no statistical reasoning. Also, the insensitivity to sample size and the hasty generalization can force us to look for hidden reasons behind seemingly non-random patterns. You have probably heard all sorts of stories about biased roulette wheels, unfair dice, and cheating croupiers that throw the ball in a special way to make all players lose. It’s true that a wheel may not be completely perfect. However, you need a very large sample to define a true pattern. Also, it’s silly to believe that a dealer can throw a roulette ball with a defined force, angle, and timing, to achieve a desired outcome.

The Law of Averages – an Insult to Knowledge

We already mentioned one fallacy that’s deceptively named as a law – the law of small numbers. There’s another one, called the law of averages. It results from poor understanding of uncertainty and probability and is tightly related to the gambler’s fallacy. The law of averages is the erroneous belief that a certain outcome is due because it is statistically possible and hasn’t occurred in recent results. Players expect outcomes to even immediately after a recent deviation from the average. This is more of a wishful thinking than a statistically adequate expectation. Once again, we must point out that the probability of a given outcome in future trials is not affected by results in the past. Keep this in mind, otherwise the law of averages will become the law of large losses.

Conclusion – How to Play Smart?

In a short summary, we must mention again that there’s more to gambling than knowing the odds if you don’t want to be a part of the world’s greatest DFS loser punishments, for example. Players often misinterpret numbers because of their biases and the tendency to use heuristics in reasoning. This escalates quickly as gamblers fall victims to outright fallacies. Therefore, the best advice is to be prepared, to know your facts, and to remember that your own mind can trick you. This way you’ll be able to avoid the traps and remain sharp. Having decent knowledge in probability and the law of large numbers will help you make more adequate decisions under uncertainty.