Gambling Mathematics – From Game Design to Strategies

Author: Dr. Catalin Barboianu

Some of us left mathematics behind long ago after graduating from school and not necessarily with regrets. Mathematics was and is the bugaboo of the school for those who have no inclination towards it. But, in everyday life, we meet mathematics at almost every action we take, and this also happens in gambling.

If you tell a non-mathematically inclined gambler what the true odds of winning a certain bet are, or how much they are expected to lose with a certain bet in the long run, or that their strategy or choices are not mathematically objective, you make both a bad thing and a good thing. The bad thing is that you take away from that person a big part of their joy of gambling; there is always a joy of gambling and it should remain so. The good thing is that you can trigger that gambler’s critical thinking and make them to want to know more about what is behind the games they play.

Mathematics stands behind gambling from the very conception of the games and plays essential roles in the gambling industry and in the players’ gambling behavior. Let’s see what the main roles of mathematics in gambling are and how you can benefit by the mathematical dimension of gambling in your play.

The Role of Mathematics in the Design of the Games

Gambling can be traced back to antiquity and forms of organized gambling emerged in the 16th century (horse racing and lotteries, especially in England). Then, gambling in casinos has been widespread from the 18th century to nowadays. All such organized forms of gambling were and are commercial endeavors, that is, forms of trade in which the organizer aims at making a guaranteed profit. But who or what is going to provide the guarantee that a game based on bets and chance will be profitable for the house or operator, given the uncertainty element? The only answer is: mathematics.

Games of chance are conceived as based on mathematical criteria. When the inventor of roulette (mathematician Blaise Pascal) designed the game in its technical side, the next question he asked for completing the design was what exactly the payout rate of each bet should be for the operator of the game to be sure that they will make a profit in the long run, whatever the bets following to be placed, and also what will make players find the game attractive? To find that, it reverts to solve a mathematical problem.

The current payout schedule of the roulette reflects those aims. If the roulette had 36 : 1 payout rate for a straight-up bet instead on 35 : 1, the house’s profit would no longer be ensured. Of if it had 36 numbers and the same payout rates.

The choices that the producers make for the characteristics of their games are based on mathematical models and applications. Ensuring the operator’s profit is done by verifying that certain statistical indicators have the desired values. It’s about notions from probability theory and statistics, like probability, expected value, and house edge.

House’s profit in the long run is expressed by the so-called house edge, a statistical average which has to be positive for any bet for the house to profit in the long run. The parametric configuration chosen for a game in the process of being conceived should obey this condition.

This is true for every casino game, not only roulette – blackjack, baccarat, craps, and any version of these, along with slots. There are the cold mathematical models behind the nice colored and sparkling games that govern their functionality and drive both the house’s profits and players’ winnings.

Mathematics and Diversity of the Games

The mathematical models behind the games also make it possible for a producer to change the parametric configuration of a game and its payout schedule to obtain new profitable versions of that game. Gamblers like diversity, like in any market customers like to be offered a wide range of products to choose from.

The widest range of diversity across all casino games was attained by the game of slots, the most popular casino game in present days. It’s not only about their theme and physical design, but more about their inner characteristics – how the outcomes are displayed (in lines or grids), the number of reels, how many paylines are available and their shapes, how many symbols appear on the reels, and so on.

Slots diversity is possible because the parametric configuration of this game holds the highest number of parameters among the casino games – the number of different symbols, the number of reels, the symbol weighting of each reel, and the number of paylines – and the possibilities of choice for these parameters are in huge number. Any particular choice of the parameters determines a new game of slots.

Unlike in the other casino games, the slots parametric configuration is not transparent, which of course prevents players or experts to evaluate the game statistically, especially in what concerns the odds of winning.

Illusions and Fairness by Mathematical Work

The mathematical work for the slots design is challenging and hard compared to other games. The games mathematician has to meet the producer’s expectations and goals by finding the right parametric choices and testing them.

Another feature of modern slots, which is also the result of mathematical design, is the false (artificial) near-misses. By manipulating the weightings of the symbols on the reels and by arrangement techniques, a machine can be set to produce frequent outcomes looking like near-misses, which encourage players to bet more. It’s the second downside of slots besides non-transparency and mathematics is responsible for it.

Mathematics is also involved in the fairness of the modern games, namely the Random Number Generators (RNG), whose algorithms are based on computable functions with special mathematical properties that ensure the uniformity and independence of the sequence of outcomes, along with other features, resulting in a “sufficiently fair” randomness, so necessary for the functioning of the games.

Mathematical Information and Responsible Play

Gambling mathematics serves not only the gambling industry, but also the players (even though not all players are aware of this role of mathematics).

First, since any game is characterized by its parametric configuration and as such is mathematically describable, getting informed about its configuration means to know the game from its essence. And it’s more than that. On the basis of a game’s mathematical parameters, one can compute the probability of winning any bet (except for non-transparent slots), and other statistical indicators as statistical averages (a bet’s expected value, RTP, house edge, variance, etc.).

Such indicators give players an objective view on how the outcomes will be distributed on average and what they can expect to win or lose over the long run with their bets. The statistical indicators are also objective criteria for the player to choose one game or another and also to choose their bets.

Today, getting informed about the mathematical facts of a game is a responsible gambling requirement. Like for any other commercial product, you should only buy it after you know its characteristics, and in gambling such characteristics are mathematical.

Gambling and Strategy

Mathematics does not only describe the games, but particular styles of gambling, and can characterize your own play in one or more games, one or more sessions, in terms of statistics. Therefore, mathematics has a role in the strategic play.

Mathematics can provide what is called an optimal strategy. An optimal strategy is a set of actions that a player can take in a game, depending on every possible situation, which ensures the best possible advantage of the player over the house or over other opponents (in poker, for instance). This advantage should be meant in terms of comparison, that is, it is better than any other advantage obtained by another strategy.

The advantage is measured in statistical terms, such as probabilities and expected values. In an optimal strategy, every move is associated with the highest possible probability or expected value and as such an optimal strategy minimizes loss and maximizes profit in the long run, but can also provide immediate better odds of winning for the next move or bet.

As a result, an optimal strategy reduces the house edge. An optimal strategy uses mathematical results from game theory, probability theory, and combinatorics. However, no optimal strategy can guarantee financial success, as the luck factor is still determinant in games of chance. It is just the best strategy you can adopt, but not every game allows such a strategy. Examples of optimal strategies are the so-called basic strategy and card counting strategy in blackjack.

But even players of games that do not allow optimal strategies, such as roulette, can benefit from the mathematics behind the games. This is true because mathematics has a role in organizing our play – analyzing bets by mathematical criteria such as probability, expected value, profit rate, and possible loss, will optimize your play with respect to aligning your financial goals, bankroll, and playing style. Being organized is always a good thing, especially in gambling, where it may prevent you from playing chaotically or in excess.

The Rational attitude about gambling

It is known that, in general, mathematical thinking contributes to one’s critical thinking, objective choices and rational actions with objective expectations. All the more, this is true in gambling, where people can easily fall prey to misconceptions and fallacious beliefs that can affect negatively their playing behavior and eventually their daily life.

Knowing as much as possible about the mathematics behind gambling will help one stay on track of responsible gambling with a rational attitude. However, mathematics is not always easy to digest, especially for those with no mathematical background, and a poor understanding or misinterpretation of the mathematical facts behind gambling may be as damaging as the lack of mathematical knowledge.

Just to take an iconic example, the Gambler’s Fallacy stems for such a misinterpretation of mathematics applied in gambling and not necessarily from not knowing much about that mathematics. Gambling can be a tricky field for those unaware, but probability theory may be as well, when applied in real life. This is why the way gambling mathematics is formulated and presented in available resources as applying in gambling is crucial for the adequate understanding of the real phenomenon.

Final Words

In conclusion, mathematics cannot be separated from gambling, as it is constitutive for the games of chance, which would not exist without their mathematical constitution.

Knowing the mathematical information about games and gambling falls within the ethical side of gaming, but is also of real help in guiding players in both their strategic endeavors and keeping them in the safe zone when gambling. Strategy based on mathematical knowledge and skills is usually successful and this is mostly visible in poker. In casino games, playing when mathematically informed has as result a rational attitude about gambling, which includes having reasonable objective expectations from the play.

But how being informed about the mathematical facts of the games and having an adequate understanding and interpretations of the mathematical facts of gambling attainable concretely? You don’t have to go back to school to learn statistics and how mathematics applies to gambling. Those who struggled with mathematics in high school and also others will not accept that as an option. Fortunately, expert resources exist on the internet that are helpful in this respect. You just have to pick the good ones. In the Casinoreviews.net blog section, you will find precise and detailed explanations of how the math behind the games is working, written by experts, for almost all casino games.