Informative-Ethical Contributions of Mathematics to Problem Gambling

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The mathematical facts of the games as information
Since games are developed around their mathematical models, these models actually define the games in the realm of gambling. There are two types of mathematical facts that define a game of chance: its parametric configuration and its probabilistic & statistical results. The first type defines the game because it is indispensable to its creation and the second because it involves the only mathematical concepts related to its “chance” feature (namely, to generate outcomes under uncertainty).
Parametric configuration is a basic numerical assignment for the game’s components participating in game’s functioning and rules and is established by game’s producer. It is also part of the mathematical models as a basic hypothesis. For example:
- There are 52 cards in a regular card deck, of which 13 values (from 2 to A) and 4 symbols (clubs, hearts, spades, diamond), each value being paired with each of the 4 symbols on the cards. This is part of the parametric configuration of a card game; 52, 13, 4 are parameters defining the game. Other parameters add to this configuration when considering a specific card game, with specific rules (for instance, the number of cards to be discarded in draw poker, the number of community cards in Texas Hold’em poker, etc.).
- There are 2 dices, each having 6 numbers on its faces in craps. Certain sums or numbers stand for the winning rules. All these numbers form the parametric configuration of the craps game.
- In slots, we have a parametric configuration of the reels, consisting in the number of reels, number of distinct symbols, number of stops on each reel and the symbol distribution on each reel; we also have a parametric configuration of the display, consisting in the topologic arrangement of the lines and paylines, the length of the paylines, etc. See more details on slots parametric configuration here.
The probabilistic & statistical results are mathematical facts derived through applications within the mathematical models and are grounded on the parametric configuration of the game. These include odds/probabilities attached to the various gaming events and statistical indicators such as expected value, house edge, volatility index, etc. Our sections on gambling mathematics in this website provide a good amount of essential probability results.
In the virtue of their defining feature explained at the beginning of this paragraph, both the parametric configuration and the probabilistic & statistical results can be considered as both knowledge and information (There is a distinction between the two terms (Turban & al., 2004).)
The exposure of the mathematical facts of a game as an ethical obligation
Exposing the parametric configuration of a game to the player prior to playing is an ethical obligation in two aspects – one commercial and the other humanitarian (Barboianu, 2013).
The commercial aspect treats the game as any commercial service, for which full technical specifications are required from the producer to the customer; a bet is still a purchased service once the player inserts a non-returnable coin in the machine.
The humanitarian aspect is related first to the free will of thought and second, to the limitation of the risk factors through further improved knowledge. Being informed on all parameters of a game one plays is a condition for unconstrained personal thinking leading to personal actions. It is as if someone asks you to bet you can jump from a high place and land on your feet; of course, if you know in advance the height from which you will jump, or measure it before you bet, you might decline the bet or propose another one for a certain measurement, and this means free decision.
Regarding the limitation of risk factors through further improved knowledge, this knowledge may be acquired either as pre-calculated numerical results such as winning odds and other statistical indicators, or by learning theoretical and applied probability theory basics. The latter variant relates directly to the cognitive contributions of mathematics in problem gambling.
Where to draw the line between ethically required and optional information?
The ethical obligation being established, the question arises as to whether this obligation should remain simply the parametric configuration of the game or be extended to include basic or advanced mathematical results coming from applications worked out on the mathematical model of that game. The extension could consist of basic pre-calculated numerical results, such as probabilities of the basic winning events and expected value, or stretch further to more complex gaming events and other statistical indicators, and the interpretation of these results. The latter variant already assumes a new level of mathematical knowledge, attainable only through didactical intervention. For the parametric-configuration-only variant, which is merely informative and either provided by the game producer or retrieved by third parties, it would remain for the player to inquire further for the mathematical results as an optional action. The question, then, clearly becomes where to draw the line between ethically required and optional information on the mathematical facts of games of chance.
On the entire range of mathematical information possible to be exposed, as the amount of information increases, there are three specific levels as seen in the next figure: parametric configuration, basic numerical results (odds of winning and EV), and knowledge of the mathematics of gambling presented in a specific teaching module.
Figure. Dividing the range of the mathematical information for a game of chance between ethically required and optional information.


If assigning the two aspects – commercial and humanitarian – of the ethical obligation over the range of the mathematical information, the commercial one covers only interval I, while the humanitarian aspect could stretch theoretically to the endpoint of interval II, if enhanced mathematical knowledge can have an impact on gambling behavior.
At first glance, the best option seems to lean toward position 2, which also has the highest number of corresponding examples from other domains; however, further interdisciplinary research is necessary for a rigorous standard, including how "best" should be defined in this particular ethical context.
The slots case
Slot machines gained and maintained this popularity despite some specific elements that could limit their appeal:
a) non-transparency: Players do not know the configurations of the machines they play at, as this information is not exposed. Blackjack players know the composition of the decks in play, roulette players know the numbers on the wheel, lottery players know the numbers from which the winning line is drawn, and so on. Slots remains the only game in which players are not aware of the essential parameters of the game, such as number of stops of the reels, number of symbols and their distribution on the reels. For more on slots parametric configurations, click here.
b) prevention from odds estimation: Obviously, the lack of data regarding the configuration of a machine prevents people from computing the odds of winning and other mathematical indicators.
The so-called PAR sheets (Probability Accounting Reports), exposing the weighting of the reels, some of the probabilities associated with the winning combinations, and other statistical indicators, are kept secret by game producers.
Recent research (Barboianu, 2014) showed that there is no rational justification for this secrecy by giving two reasons, one psychological and the other mathematical. For the latter, mathematics provides us with some statistical methods of retrieving the missing data, which are essential for the numerical probability computations in slots. Click here for an overview on these statistical methods and the project Probability Sheet for any Slot Game. For more on slots mathematics, click here.
Resources for the information
The information required to be exposed as parametric configurations would be in the form of a technical/mathematical sheet specific to each game, consisting of those parameters of the mathematical design of that game that define the sets of possible outcomes and are essential toward probability and statistical computations. For example, in slots the parametric-configuration sheet must show the number of distinct symbols, number of stops of each reel, and the symbol distribution (weighting) of each reel. In a card game, the number of decks used, the number of cards in each deck, and the composition of each deck (numbers of card values and symbols) are known. With a drawing machine (for example, lottery or bingo), the total number of numbers/balls, their value interval, the number of numbers/balls to be drawn, and so on are likewise known.
The gambling mathematics knowledge may be acquired by gamblers via instructional means (gambling-mathematics courses in schools or private organizations, experimental interventions) or specific media (books, journals, magazines, and websites). Of course, the structure and content of such gambling-mathematics resources vary and the existent courses usually follow the curricula of regular Introduction to/Basics of Probability and Statistics classes in post-secondary schools, with the focus on applications of these disciplines in gambling. Regarding the latter category of resources, the plethora of popular literature on gambling mathematics published in the last decade raises the necessity of critical selection and professional certification when it comes to a recommendation. This is because such commercial publications serve various scopes, wide or narrow, and the information delivered by them can be useful even if  incomplete, but also misleading (as is the case with most of the so-called “how to win” or “strategy” titles, in which the mathematical information or systems described are not mathematical at all). In any case, credentials of the authors and publishers of such publications must be verified.
This website provides you with basic information on gambling mathematics for the most popular games (especially probability results).
Further research is needed to establish the optimal content and organization of the basic mathematical information with respect to the ethical obligation and whether it can be comprised in the form of warning messages similar to the warning messages used in the prevention of other addictions. Projects forthcoming such research may be devoted to the development of the optimal informative modules and the technical ways of exposing them.



Barboianu, C. (2013). The mathematical facts of games of chance between exposure, teaching, and contribution to cognitive therapies: Principles of an optimal mathematical intervention for responsible gambling. Romanian Journal of Experimental Applied Psychology, 4(3), pp. 25-40.

Barboianu, C. (2014). Is the secrecy of the parametric configuration of slot machines rationally justified? The exposure of the mathematical facts of games of chance as an ethical obligation. Journal of Gambling Issues, 29, pp.1-23.

Turban, E., Aronson, J. E. & Liang, T.P. (2004). Management support systems : an overview. Decision Support Systems & Intelligent Systems, seventh ed., pp.2-23. Upper Sadle River: Prentice Hall.


 This entry should be cited as:

Barboianu, C. (2014). Informative-Ethical Contributions of Mathematics to Problem Gambling. Retrieved from


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The author of this page is Catalin Barboianu (PhD). Catalin is a games mathematician and problem gambling researcher, science writer and consultant for the mathematical aspects of gambling for the gaming industry and problem-gambling institutions.

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