Mathematics of Gambling

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People engage in casino gaming by wagering or staking money because it is exciting, and there is a chance of winning a prize. The fun and anticipation make it a popular event, with individuals relating gambling success to luck. However, its prosperity depends on mathematics. Development of games revolves around mathematical models that define its association with gambling. Two mathematical facts determine a game of chance: its parametric arrangement, which is the basic numerical configuration of the game’s components, and its results, which depend on probability and statistics. Three basic mathematical principles underlie gambling: probability model, expected value, and volatility index. These concepts help us to understand how casino games work and how people use this knowledge to increase their chance of winning.

Probability Model

The mathematics of games is a collection of probabilities in a game of chance. In essence, the model starts with an experiment and generates the mathematical structures of the field of events, which are the primary units of probability theory. Therefore, the probability is the chance that a given situation will yield a particular outcome.

There are three ways of expressing the probability theory: as a fraction, as a percentage, odds, or decimal. All chances are numerical, and they exist between 0, which is an impossible outcome and 1, which is a certainty (De Finetti 25). Probability in gambling consists of the following events:

Events that relate to the player’s or opponent’s play.

Events associating an individual play or to another person’s chance.

Long-shot or next events.

The first principle to understand about this theory is that it applies to random events. Its application begins with the comprehension of sample space, which is a description of all possible outcomes; for instance, the probability of rolling a six-sided dice has six possible results: 1, 2, 3, 4, 5, or 6. The sample chance is six, and the likelihood of any specific side to play is 1/6. In this model, the terms “OR” and “AND” means add and multiply respectively (Gillies 9). For this example, the probability of getting either 1 OR 2 is the sum of their unique occurrence, giving a fractional outcome of 2/6 as shown in Equation 1 below.


Whereby 1/6 is the probability of getting a 1 and 1/6 is the likelihood of getting a 2.

In the equation above, the results show probability as a fraction. Expressing the outcome as a percentage gives 33.33%, as a decimal it gives 0.33, and the odds chance is 2 to 1. Therefore, the probability is a mathematical principle that describes the opportunity of a specific outcome of a game.

Expected Value

Professional gamblers are not only interested in probabilities, but also in the amount of money they can theoretically win from a game. This concept is known as the expected value (EV), and it is the summation of the likelihood of all independent outcomes multiplied by the payoff as shown by Equation 2 below. Therefore, this model represents the amount a player can win per bet for multiple wagers.


Whereby X

is the discrete random variable,  are the corresponding values,  is the probabilities, and  is the sum of series.

Consider a game of American roulette. In this event, there are 38 possible outcomes. Placing a wager on a single number pays 35 times, and your bet returns: you get 36 times your chance. Setting 1 dollar severally on a unique number gives an EV of -$0.05 shown by Equation 3 below; thus, the player will be losing five cents for every one-dollar stake.


The expected value is a weighted mean, representing the amount a player should get by repeating the gaming situation severally (Samuelson. Moreover, the model incorporates identical conditions and applies mathematical probability by placing the same type of stake. A negative value means that the casino benefits, while a positive EV implies that the player wins the bet. Therefore, this mathematical principle plays a critical role in making decisions in games and betting for stakes with specific payouts.

Volatility Index

The volatility index is a statistical theory that uses standard deviation (SD) to predict the variation between the actual and expected win percentage value for a given number of wagers. In this model, SD determines the magnitude of the outcomes, its fluctuation with theoretical figures, and the fluctuation (Ferrari). Further, this principle incorporates the limit theory, resulting in two volatility analysis guidelines: it is only 5 % of the whole event will the outcome be more than two standard deviations from the expected value, and only 0.3% (almost never) of the time will the result be more than three SD’s from theoretical outcome. Equation 4 shows SD.


Where N = the number of values evaluated, x = each value in N, μ = the mean, and Σ is the sum of parameters from the first value (I = 1) to the last value (N).

To understand volatility principle, consider the casino game of craps. Using a series of 1,000 wagers and 1.4% house advantage, on average the player will be 14 units behind in each event. For a single bet, the SD is 1.0, while for 1000 it is 31.6. From the volatility analysis guidelines, there is a 95% chance that the gambler’s real win will be 49 units ahead and 77 units behind, and almost certainly between 81 and 109 units ahead and behind respectively. Therefore, this model uses a statistical measure to determine the deviation of the actual outcome from the expected value.

Application of Mathematical Principles in Gambling

It is essential to understand the mathematics behind the game of chance to choose the most appropriate and sensitive wagering strategy. People in the gambling business use mathematics principles to increase their betting skills. Firstly, they understand that the theory of probability forms the basis for every game. It is this concept that relates to the chances of reaching a particular outcome, while dependent and independent events are critical in determining the betting amount and the most appropriate time to stake a wager (Wackerly et al. 40). For example, in a five-card poker, the chance of withdrawing four similar samples is 0.000240, while the probability of getting a royal flush is 0.00000154 that is the rarest hand. Skilled gamblers recognize the sample spaces for each game and the chances that relate to each side. As a result, they can estimate the odds of a specific hand, guiding their betting choice.

Secondly, gamblers use the concept of expected value to increase their skills. This principle informs them how much money they will lose or win in a game. For instance, if a player is paid $1 for each time the dealer flips head and a deduction of a similar amount if the outcome is tails, the EV would be zero because the probability of heads is occurring equals to that of tails as shown by Equation 5below. 


In this case, the game is said to be “fair” because the game has no monetary advantages or disadvantages irrespective of the number of events. However, if the dealer pays $1.50 each time the player flips head, then the EV is $0.25 as shown in Equation 6 below. Playing the game, a hundred times yield an outcome of $25. Thus, gamblers use this concept to establish the how much they will gain in the long run.

                         EV = 0.5$1.50 +0.5- $1.00 = $0.25                              (6)

Thirdly, professional gamblers depend on the volatility index to increase their chance of winning. They use this principle to determine their chances of winning more or less. For instance, in the above coin scenario, the player has a 68% probability of earning -$10 and 10$ after 100 events. Additionally, there is a 95% chance of leaving the game -$20 and 20$. From the volatility index, people in the gambling business can establish their odds of earning more than the expected value for a particular round. Games that are highly volatile have a wider variation between the expected and actual results than those with low tendencies, giving gamblers a greater chance of winning above the EV. As a result, this mathematical principle helps players to quantify their luck.


Generally, games incorporate mathematical facts such as parametric configuration, probability, and statistics. Gaming events have an association with gambling. There are three principles behind a betting: probability model that determines the likelihood of each outcome, the expected value that identifies the number of gains or loss in each game, and the volatility index that compares the variation between the actual and theoretical value. Skilled gamblers rely on these mathematical concepts to assess the risk around an event by analyzing properties relating to probability, winning odds, expected value, and volatility index. Therefore, mathematics is a crucial factor in games and gambling.


De Finetti, Bruno. Theory of probability: a critical introductory treatment. Vol. 6. John Wiley & Sons, 2017.

Ferrari, Mario Anthony. Slot machine gambling and testosterone: evidence for a ‘winner-loser’effect?. Diss. University of British Columbia, 2017.

Gillies, Donald. Philosophical theories of probability. Routledge, 2012.

Samuelson, Paul A. "The “fallacy” of maximizing the geometric mean in long sequences of investing or gambling." THE KELLY CAPITAL GROWTH INVESTMENT CRITERION: THEORY and PRACTICE. 2011. 487-490.

Wackerly, Dennis, William Mendenhall, and Richard L. Scheaffer. Mathematical statistics with applications. Cengage Learning, 2014.

September 25, 2023


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Casino Gambling

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