It is often said that **gambling math skills** are the decisive factor that determines one’s transition from an average gambler to a professional one and gets them on the edge of success. But what exactly are those skills? Do we have to go “back to school” to achieve them? Are they achievable by anyone, regardless of their mathematical background? In this article, I will answer these questions and also explain to what extent such skills contribute to the monetary success of gambling.

*TOP*

## Gambling math

**Games of chance are conceived on the basis of precise mathematical models**, whether we talk about casino games like roulette, blackjack, and so on, or sports betting. These models ensure that the producers and the operators of their games are functional and that they won’t go bankrupt with them in the long run. Without this mathematical grounding, such games would never run. This math-indispensability principle also determines the gamblers’ behaviour with respect to using gambling math in their thinking about the games and their play. “Using” here means general application of mathematics in gambling for mathematical description of the games, game analysis, evaluation of a particular play, strategic play, or optimal play.

Such application is done by means of one’s gambling math skills and obviously relies on the mathematical theories involved, which belong to fields like Probability Theory, Measure Theory, Statistics, Game Theory, Set Theory, Algebra, and Combinatorics.

## Back to school?

The above names may sound scary for one with elementary or no mathematical background and the truth is that such fields are complex and most of them are studied in colleges or universities with mathematical profile. Should this fact make the average gambler getting resigned regarding the acquisition of the math skills required for moving to a professional status? There is good news and possibly bad news.

The good news is that the above mathematical fields can be narrowed to precise notions, results, and application facts that reflect the application of their mathematical theories in gambling, and professional resources are available *(in print and on the internet)* **from which you can learn how to apply math in gambling**. In other words, you don’t have to study all the mentioned fields of mathematics, only what is relevant and conclusive with respect to the proposed aim, so going “back to school” is not an effective option.

The bad news *(if it’s bad indeed)* is that **understanding, and adequate interpretation of the learned notions and results are essential for the correct application of gambling math**; such skills are not taught in math classes and are dependent on everybody’s innate cognitive profile.

## Skills beyond mathematical

Probability theory and statistical notions can be very tricky for those unfamiliar with them and even for math-inclined people. Besides deep understanding, they require an adequate interpretation when applied in the real world. It happens that many of the so-called gambling cognitive distortions *(like The Gambler’s Fallacy, Conjunction Fallacy, and so on)*, which are objects of investigation in problem gambling, stem not necessarily from the lack of mathematical knowledge but rather from the inadequate perception, understanding, and interpretation of the notions in the real world of gambling.

Probability theory employs primitive notions – such as that of **randomness** – which are not mathematical. No mathematical course or class will teach you about the nature of randomness. This is another reason why going back to school to learn about mathematics applicable to gambling is not effective – because the application of mathematics in gambling goes beyond formal mathematics. Knowing the mathematical definition of expected value will not guarantee one’s understanding of how this notion gets materialized in one’s play. To give an isolated *(but representative)* example, knowing the definition of the RTP *(Return to Player)* will not answer a slot player’s questions related to common misconceptions about the RTP – for instance: Does this percentage apply to one player’s wagering or all players’ spinning that machine? Does it apply for a certain or any number of spins?

Knowing and understanding basic notions of gambling math, such as probability, expected value, and the house edge, is essential for gambling math skills. However, their adequate interpretation goes beyond mathematical skills and depends on everybody’s cognitive assets and profile. In our examples, the correct understanding and adequate interpretation of the notion of statistical average complete the mathematical knowledge about it. Yet there are experts who can assist you with such matters, and there are resources available on such topics for you to consult.

## Gambling math skills

As explained in the previous section, **gambling math skills go beyond the knowledge of the formal mathematics of gambling**. This is why we can categorize them as **educational skills** and **cognitive skills**. The latter skills are associated with gambling math knowledge when it comes to its application in real play.

### Educational skills

When enumerating the skills, it is justified to start with the ability to **get informed** about a game before playing it. The characteristics of a game are determined by the mathematical parameters that define it *(the structure of the outcomes, probabilities associated with the outcomes, payout odds, expectations, and other types of statistical averages)*. This **educational skill is actually a norm of responsible gambling and assumes understanding those parameters in their mathematical nature and interpreting them adequately**. The mathematical characteristics of a game stand as objective criteria for choosing a game and also for building strategies for playing it.

**Knowing the basics of gambling math** *(such as the formulas for combinations, the definition of classical probability, basic properties of probability, probability as a measure, expected value, variance, the Law of Large Numbers, and so on)* is the main educational skill, which can be acquired by studying resources or attending a course. This general skill assumes a good mathematical understanding of the notions and results and how they are applied in gambling.

### Cognitive skills

Knowing the math behind gambling is not all when it comes to applying it, as in real play, we are not in the circumstances of math homework being done in the comfort of our home or class. We gamble under the pressure of time and house’s or casino’s rules – decisions have to be taken in a very short time and we are not allowed to consult any materials while gambling. The exception to this principle is sports betting, where we choose and build our bets in advance.

The cognitive skills related to the math of gambling, required for games where decisions have to be made during the game, are **memorization, quick mental calculation, and approximation**. Memorization is extremely important for optimal play based on complex strategy charts that need to be known beforehand *(applicable in blackjack and circumstantially in baccarat)*. Quick mental calculation and approximation are very useful for either expectation – or probability-based strategies applied in combinatorial games *(especially in card games like poker)*, where the optimal decision depends on estimating and comparing probabilities that cannot be computed precisely during the live game. Quick mental calculation is also required in the card-counting strategy. It is worth noting that approximation is a mix of educational and cognitive skills since the process is based on pure mathematics.

A related underlying cognitive skill is **distributive observation**, which is used in card games where you need to quickly observe and track cards in your hand and your opponent’s *(or dealer’s)* hand.

## Games in which gambling math skills are applicable

Gambling math skills, either educational or cognitive, are actually **applicable in all games of chance**.

Even playing games based on pure luck, such as roulette, craps, or slots, can benefit from such skills. This is true because **mathematics also has an organizing role and not only a strategic role**. Knowing in advance the probabilities associated with the various bets and other statistical or financial indicators of them *(an educational skill)* can help in choosing the suitable games and/or bets or betting schemes relative to your personal criteria of playing and goals. The same stands for sports betting, where choosing and organizing are the main components of professional betting. In addition, math-based strategies in betting, such as the Kelly Criterion, require knowing that formula and applying it concretely.

In blackjack, applying a card-counting strategy exploits to the maximum the cognitive skills of memorization, quick mental calculation, and distributive observation. In the circumstance of a baccarat table with several players, the same skills are required for card counting *(although this strategy is not as effective in reality as it is in blackjack)*.

Not many people know that in slots mathematics can have organizing and strategic roles. While it is well known that the parameters of the slots games are kept secret by their producers *(hence no probability computation can be made)*, one can retrieve by statistical methods based on tracking and recording indicators like RTP *(if not provided)* and volatility and also approximated probabilities of winning for the prizes. Moreover, there is a particular situation where optimal play can be applied in slots, namely when chasing a progressive jackpot over the long run in must-hit-by progressive slots. In this situation, the breakeven point *(point with zero expectation)* can be determined by applying a particular formula, and the strategic advice is to bet only when the level meter passes that point.

In poker, all educational and cognitive skills are in high demand, and the ability to categorize and approximate probabilities *(or recall them from memorized tables)* in a very short time is decisive for deciding the right move.

## Achieving and training gambling math skills

**The only way one can achieve their educational skills is by self-study or learning with an instructor.**The former option requires educational resources, and this raises the issue of selecting such resources, especially when talking about web resources. The expertise and credentials of the authors of the web resources

*(and even written materials such as books)*should be checked before using their products, as teaching gambling math is not in everybody’s hands. We have already seen how important it is to complete the gambling math knowledge with knowledge about the adequate interpretation of the mathematical facts of gambling. The latter knowledge should be delivered only by experts with multidisciplinary expertise, including but not limited to mathematics

*(cognitive psychology and philosophy of mathematics are disciplines involved)*. When choosing to attend courses instead of self-studying resources, verification for credentials should be easier, especially if the authors or instructors deliver the course within a recognized educational institution.

Achieving and consolidating gambling math educational skills requires dedication, time, and effort, preferably before throwing yourself into gambling for the long term. While the effectiveness of the acquired knowledge is mainly dependent on the quality of the curricular content and the methods of the instructor, we must say that such an educational task cannot be effectively fulfilled by any student. The mathematical background of the student counts; moreover, some people are inclined to math, and others are not. As such, these educational skills rely more or less on the innate cognitive features of the students.

As for the gambling math cognitive skills, the same factor of innateness applies. As applied mathematics can be trained by long-term exercise, these cognitive skills can be trained as well. The difference between the two is that, while educational skills are trained “offline”, the cognitive skills can only be trained during the perseverant gambling practice or simulations of the real play.

For both categories of skills, the first recommendation for training them is to never give them up, to stick with the strategies based on them, and resist the temptation to drop them for subjective or illusory feelings, even if this means ignoring the unfavourable outcomes of the games for a long period.

## Gambling math skills and success in gambling

Gambling math is usually associated with success in gambling more in terms of professional gambling and less of tracked record. But what makes a player successful? Its proven record of monetary success or their gambling math expertise?

**No strategy, either based on mathematics or of another kind, guarantees monetary success in gambling.** If it were otherwise, we would see everywhere mathematicians getting rich and leaving their jobs for gambling. The explanation of this principle resides in the nature of gambling as a random experiment and can be mathematically proved within probability theory.

Betting systems such as the Martingale, for instance, although based on mathematical truths, cannot guarantee profit. This is because a long streak of losses to deplete your bankroll is possible at any time, and moreover, over the long run, the expected gain is the same as the expectation of a static or flat bet, given by a constant house edge.

Optimal play, which is a mathematics-based strategy of maximizing profit and minimizing loss by using the available information in the game, does not guarantee success in either the short or long term. It only does what its name says, that is, **optimizing**. In other words, it is better to use it than not in the long run, but the “better” is only meant as ‘theoretically better’; that is, no other strategy will offer a higher expectation.

In front of these ‘actually pessimistic’ aspects of the gambling math, one may fairly ask **why we need such skills anyway if they do not guarantee winning and success**.

The answer is not straightforward. An optimized expected value can materialize in the real play sooner, later, or never after 100, 1,000 or 1 million spins. As well, a move with a higher probability of winning may lead to a win immediately or when repeated after hundreds of plays. Yet, **using your gambling math skills makes a difference**. It is not essential that the difference may occur late, but just that it is the right move, the best you can do.

## Proven success in using gambling math skills

The father of the card-counting strategy, mathematician Edward Thorpe, was banned across the US casinos in the 1960s’. More recently, mathematician Michael Shackleford declared that he was banned from some casinos just for his gambling math expertise. **Is there a justified reason why casinos are afraid of games mathematicians? Yes and no.** No, because the house will have an overall profit whatever strategy their players use, and no, again, from an ethical perspective. Yes, because this profit may be slightly lower if skilled players are involved.

Our examples were from blackjack, where the optimal strategies are the most effective *(in the long run)*. **As for a game where all the gambling math skills contribute to success more than in others, this is poker.** There are two main reasons for that:

1. Poker is a game where the expected value of a bet gets materialized quicker than in other games – the factors causing that are the brevity of a round, leading to a high number of rounds per session, and the high frequency of repeated categories of card configurations for the board.

2. In poker, most of the card probabilities are very difficult to compute or approximate and are in a number of millions, even representing categories of similar configurations. Those who have the ability and skills to estimate and approximate them and use the results strategically during the game are gifted and get on edge. It is not surprising at all that the most mediatized successful poker players are constant winners and declare that they have mastered the math of poker *(even though other skills specific to poker – such as psychological skills – are responsible as well for their success)*.

## Conclusions

Gambling math skills are abilities that involve both knowledge about the mathematics of games and gambling and cognitive skills associated with this knowledge. Not only a deep understanding of the mathematical notions and facts of gambling are required for such skills to be effective, but also their correct application and adequate interpretation in real play. This is why achieving gambling math skills does not necessarily require attending courses of formal mathematics applicable to gambling but rather self-studying expert resources based on multidisciplinary content in the field of gambling math.

Both the educational and the cognitive skills are trainable by long practice; however both categories of skills rely on innate cognitive assets, which are different from person to person.

The gambling math skills make a difference in one’s play, in both the organizing and strategic aspects of their gambling. Although the effectiveness of using the math skills in gambling remains theoretical, it is better to use them than not, and the most relevant examples for concrete successes come from blackjack and poker.