Gambling has always been a popular pastime, but few people think about how exactly the mechanisms that determine winnings and losses work. The mathematics of gambling is a set of mathematical principles governing outcomes and probabilities in various games of chance. It determines how often players can expect to win or lose, as well as what edge the casino has in each game. Through the mathematics of gambling, it is possible to calculate how much you can win or lose on average over the long run, which is key to understanding the basic principles of how a casino works.
In this article, we will look at various mathematical concepts that are crucial to understanding how casino games work. These include probability theory, mathematical expectation, variance, and the casino edge. Understanding these concepts will help players evaluate their chances of success and make more informed decisions during the game.

Basic Concepts of Probability in Gambling

Probability theory is a branch of mathematics that studies random events and their probabilities. In the context of gambling, an understanding of probability theory is key to understanding how often you can expect to win or lose, and to assess risks and potential rewards. Let’s consider the main concepts of the theory of probability, which are applied to gambling.

Probability

Probability is a numerical expression of the possibility of the occurrence of a certain event. It is measured from 0 to 1, where 0 means the impossibility of the event, and 1 means its certainty. In the context of gambling, probability is used to estimate the chances of winning or losing in various games. Understanding probability helps players make informed decisions, choose strategies, and understand which outcomes are more likely.
Using probability allows you to predict how often a certain outcome may occur in the long run, which is especially useful for developing game strategies.
In the game of roulette, the probability of winning depends on the type of bet and the number of possible outcomes. For example, European roulette has 37 sectors (numbers 1 to 36 and one zero).

1. Bet on one number (Straight Up):
   Probability of winning: 1/37≈0.027 or 2.7%.
   This means that the chance of winning by betting on one specific number is approximately 2.7%.
2. Bet on red or black:
   Probability of winning: 18/37≈0.486 or 48.6%.
   In European roulette, there are 18 red and 18 black numbers, and one zero (green). Thus, the probability of winning when betting on red or black is 48.6%.

In a dice game (such as Craps), the probability of an outcome depends on the number of possible combinations that can occur when two dice are rolled.

1. The probability of getting a certain number (for example, 7):
   Possible combinations for number 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
   Number of possible combinations: 6.
   The total number of combinations when rolling two dice: 6×6=36
   Probability of getting 7: 6/36=1/6≈0.167 or 16.7%.
2. Probability of getting a number 2 or 12:
   Possible combinations for number 2: (1,1). For the number 12: (6,6).
   The number of possible combinations for each of the numbers: 1.
   Probability of getting 2 or 12: 1/36+1/36=2/36=1/18≈0.056 or 5.6%.

Odds

In the world of gambling, two important concepts are distinguished: probability and odds. Although both terms are often used to assess the possibility of a certain event occurring, they have different mathematical meanings and ways of expression.
Probability is a numerical value that reflects the possibility of a certain event occurring. It is measured between 0 and 1. For example, if the probability of winning a certain game is 0.25, this means that, on average, the event will occur 25% of the time, or 1 time out of 4.
The formula for calculating the probability of occurrence of event A looks like this:
P(A)=Number of favorable outcomes/Total number of possible outcomes

Odds express the ratio of the number of favorable outcomes to the number of unfavorable outcomes. It is a measure of how likely an event is to occur compared to how likely it is not to occur. Odds can be expressed as a ratio (fractional odds) or as a decimal coefficient.
The formula for calculating the odds (Odds) for the occurrence of event A:
Odds(A)=Number of favorable outcomes/Number of unfavorable outcomes.

Odds and probability are interrelated, but reflect different aspects of the occurrence of an event. Probability shows the possibility of an event occurring, while odds show the ratio between the occurrence and non-occurrence of an event.
Consider an example with a coin toss.

• Probability of falling eagle: 0.5 or 50% (1 favorable result / 2 possible results).
• Chances of getting an eagle: 1:1 (1 favorable result : 1 unfavorable result, since there are only two possible results – “eagle” or “tail”).

In this case, probability and odds help to represent the possibility of an event in different ways: probability as a percentage possibility, and odds as a ratio.

Expected Value

Expected Value (EV) is a mathematical concept used to determine the average outcome of a bet over the long run. Expected value shows how much a player will win or lose on average per bet if played many times. This is an important indicator for players, as it allows you to assess the long-term profitability or loss of various bets in gambling.

Expected value is calculated as the sum of all possible outcomes multiplied by their probabilities. The formula for calculating EV looks like this:
EV=(P(W)×W)+(P(L)×L)
also:

• P(W) is the probability of winning,
• W is a win,
• P(L) is the probability of loss,
• L is a loss (usually expressed as a negative value).

Consider an example of betting on one number in European roulette:

• European roulette has 37 sectors (numbers from 1 to 36 and one zero).
• The payout for a single number bet is 35 to 1.
• The probability of winning when betting on one number: P(W)=1/37≈0.027 or 2.7%.
• Probability of loss: P(L)=36/37≈0.973 or 97.3%.
• If the bet is $1, the winnings in case of success will be $35 (net amount without taking into account the initial bet).

Then the expected value (EV) will be: EV=(0.027×35)+(0.973×(−1))=0.945−0.973=−0.028
Therefore, the expected cost of betting on one number in roulette is -0.028 $. This means that, on average, a player loses 2.8 cents for every $1 bet over the long haul.

House Edge and Its Impact on Players

The house edge (or house edge) is the mathematical advantage the casino has over the players in any game. This advantage ensures that the casino makes a profit in the long term, even if individual players may win in the short term. The casino edge is expressed as a percentage and represents the casino’s average profit from each bet placed by the player. For example, a casino edge of 5% means that on average the casino wins 5% of all bets placed.

Blackjack

In blackjack, the casino edge can vary depending on the rules of the game, such as the number of decks used, the possibility of doubling bets and insurance.

• Basic House Edge: In standard blackjack, the house edge is usually around 0.5% if the player plays the basic strategy correctly. This means that for every 100 hryvnias bet by the player, the casino expects to make a profit of $0.50 in the long run.
• Edge Calculation: The casino edge in blackjack is calculated based on the winning probabilities of the player and the casino, taking into account the possibility of a draw and different betting options.

Roulette

Roulette has different variations (European, American), each of which has its own casino advantage.

• European Roulette: Contains 37 sectors (numbers from 1 to 36 and one zero). The advantage of the casino in European roulette is 2.7%. This means that for every 100 hryvnias bet by the player, the casino expects to make a profit of $2.70.
  Casino advantage (European roulette)=1/37×35−36/37×1=−0.027=2.7%
• American Roulette: Contains 38 sectors (additional double zero). The casino edge here increases to 5.26% as another sector is added which reduces the player’s chances of winning.

Casino advantage (American roulette)=2/38≈5.26%

Slots

Slots have different payout percentages (Return to Player, RTP), which can vary from game to game.

• Calculation of the casino edge: The casino edge for slots is calculated as 100% minus the RTP. For example, if a slot has an RTP of 96%, the house edge is 4%.

Casino advantage (slots)=100%−96%=4%

Common Mathematical Strategies in Gambling

Gambling has always attracted players with its exciting and unpredictable nature. However, behind the apparent randomness there are deep mathematical patterns that can help players make more informed decisions and increase their chances of success. Common mathematical strategies, such as basic blackjack strategy, the Martingale system, card counting, and others, allow players to better understand the mechanics of the game and estimate the probability of different outcomes. In this article, we will look at the most popular mathematical strategies used in various gambling games and discuss how they can help players minimize their risks and maximize their opportunities.

Martingale System

The Martingale betting system is one of the most famous and simplest betting strategies used in gambling games such as roulette, blackjack and others. The basic idea behind this system is for players to double their bets after each loss in the hope of winning back all previous losses and making a small profit.
How the Martingale system works

1. Starting Bet: The player starts with a small starting bet, for example $1.
2. Doubling after Losing: If a player loses, they double their bet in the next round ($2 bet).
3. Continued Doubling: The player continues to double his bet after each loss ($4, $8, $16, etc.) until he wins.
4. Return to the original bet after a win: As soon as the player wins, he is returned to the original bet ($1).

The purpose of this system is to ensure that on the first win, the player not only covers all previous losses, but also receives a profit equal to the initial bet.
Suppose a player bets on red in roulette and starts with $1:

• 1st round: bet $1, lose ($-1).
• 2nd round: bet $2, lose (-$3 total).
• 3rd round: bet $4, lose ($-7 total).
• 4th round: bet $8, win (+$8, total win: $1).

The player wins $8, which covers all previous losses (1 + 2 + 4 = $7) and brings $1 in profit.

Kelly Criterion

The Kelly Criterion is a mathematical formula used to determine the optimal bet size in gambling or investments to maximize long-term capital growth. The criterion helps players and investors find a balance between potential profit and risk, minimizing the probability of complete ruin.

Kelly’s formula is as follows:f∗=(bp−q)/b
also:

• f∗ is the optimal share of the bankroll for the bet,
• b is the winning ratio (net winnings from the bet per unit),
• p is the probability of winning,
• q is the probability of loss (1 – p).

How the Kelly test works
The Kelly Criterion allows a player to calculate how much of their bankroll they should bet on each game or investment to maximize their long-term capital growth. It takes into account both the probability of winning and the potential profit from winning.
Suppose a player has a bankroll of $1000 and considers a bet with the following parameters:

• A bet on an outcome that pays 2 to 1 (ie, b=2),
• The probability of winning this bet is 60% (p=0.6p),
• The probability of losing is 40% (q=1−p=0.4).

Advantages of using the Kelly criterion

1. Capital Growth Maximization: The Kelly Criterion allows players to maximize their long-term capital growth by calculating the optimal bet size for each game or investment.
2. Minimizing the risk of ruin: Using the Kelly criterion helps to minimize the risk of complete ruin, as the strategy automatically adjusts the bet size according to the change in probabilities and capital. If the capital decreases, the optimal rate also decreases, which reduces the risk of further losses.
3. Flexibility: The Kelly Criterion takes into account both the probability of winning and the payoff, allowing you to adapt the strategy to any conditions.

Card Counting in Blackjack

Card counting in blackjack is a mathematical strategy that allows players to gain an edge over the casino by keeping track of the ratio of high cards (tens and aces) and low cards (twos to sixes) remaining in the deck. The purpose of card counting is to determine when the odds are in a player’s favor and increase or decrease bets accordingly.
Card counting is based on the fact that high cards (tens and aces) are more beneficial to the player and low cards (twos and sixes) to the dealer. In a standard game of blackjack, a high number of high cards in the deck increases the probability of getting a “blackjack” (21 points) or winning from a dealer who can “go over” (exceed 21 points).
Players use simple scoring systems such as the Hi-Lo system where:

• High cards (10, J, Q, K, A) count as -1.
• Low cards (2, 3, 4, 5, 6) count as +1.
• Middle cards (7, 8, 9) count as 0.

Players add or subtract the value of each card that comes out of the deck to maintain a “running count”.

How card counting works

1. Card Tracking: The player keeps track of all the cards that have come out of the deck.
2. Calculation of “true score”: The current score is adjusted for the number of decks remaining in the game. This is called the “true count” and is used to estimate advantage.
3. Resize bets: When the real score is positive, the player has an advantage and can increase the bets. When the true score is negative or zero, the bets are reduced.

Probability Distributions in Gambling

Gambling is an integral part of culture and entertainment around the world, but behind each game there are complex mathematical models that determine its results. Probability distributions are fundamental concepts that allow us to understand how often different outcomes can occur in games of chance such as roulette, blackjack, poker, or slots. These distributions help players assess the odds of winning, understand potential risks and develop strategies to maximize their opportunities. In this article, we will look at the main types of probability distributions used in gambling and explain how they affect gameplay and decision making.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of ( n ) independent experiments, each of which has two possible outcomes: success (with probability ( p )) or failure (with probability ( 1 – p )).
The formula for calculating the probability of obtaining exactly (k) successes in (n) trials looks like this:
P(X=k)=(k/n​)pk(1−p)n−k
also:

• ( \binom{n}{k} ) is the binomial coefficient, which is calculated as ( \frac{n!}{k!(n – k)!} ),
• ( p ) is the probability of success in one trial,
• ( k ) is the number of successes,
• ( n ) is the total number of trials.

Application in games

1. Coin tossing: Each time a coin is tossed, there are two possible outcomes — heads or tails. If ( p = 0.5 ) for each outcome, then the binomial distribution can be used to calculate the probability of getting a certain number of heads (or tails) after several tosses.
2. Win/Loss Bets: In the case of bets where there are two possible outcomes (win or lose), the binomial distribution helps determine the probability of a certain number of wins for a given number of bets if the probability of a single bet winning is known.

Normal Distribution

Normal distribution is one of the most important and widespread probability distributions in statistics and probability theory. It is often called the Gaussian distribution and is bell-shaped, characterized by symmetry around the mean. The normal distribution is an important tool for modeling and analyzing random processes, especially in the context of games of chance with many trials.
The normal distribution is determined by two parameters:

• Mean (mathematical expectation, μ): Indicates the central value of the distribution, where most of the values ​​are concentrated.
• Standard deviation (σ): Defines the width and shape of the distribution. A larger standard deviation means more variation in the data around the mean.

The graph of a normal distribution is bell-shaped, where the highest frequency of values ​​is observed near the mean, and the probability of values ​​deviating from the mean decreases symmetrically on both sides.

Central limit theorem (CLT) is a fundamental principle of statistics that states that the sum or mean of a large number of independent random variables, even if they are not normally distributed, approaches a normal distribution as the number of trials increases.
In the context of gambling, CGT explains why the outcomes of large numbers of repeated games (such as dice rolls, roulette spins, or blackjack hands) tend to follow a normal distribution around the mean. This means that while individual results may vary greatly, the overall total after many games will be closer to the mathematical expectation.
Suppose a player plays a game with an expected mean win of $1 per game and a standard deviation of $5. According to CGT, if a player plays 1000 games, the distribution of the average winnings will approach a normal one. The average win will be $1,000 ($1 × 1,000), but it can vary by several standard deviations (approximately $5/√1,000 ≈ $0.16). This knowledge allows the player to predict his chances of winning or losing in the long run.

Poisson Distribution

The Poisson distribution is an important tool in probability theory and statistics, used to model the number of events occurring in a fixed interval of time or space. This distribution is particularly useful when events occur independently of each other and with a constant average frequency.
The main properties of the Poisson distribution:

1. Parameter (\lambda): This is the average number of events occurring in a fixed interval.
2. Probability (P(X = k)): The probability that (k) events will occur in a fixed interval.

Simulation of winning the jackpot in slot machines:

1. Fixed interval: For example, consider a certain period of time or the number of game attempts.
2. Average win rate ((\lambda)): If it is known that the jackpot is won on average once in 10,000 attempts, then (\lambda = 1/10,000).
3. Probability of winning: Using the Poisson distribution formula, you can calculate the probability of winning the jackpot a certain number of times for a given number of tries.

Risk of Ruin in Gambling

The risk of going broke (or the risk of losing) in gambling is the probability that a player will lose their entire bankroll during a game. This concept is important for all players, regardless of their experience or chosen strategy, because it determines how likely a player is to run out of funds to continue playing. The risk of ruin is defined as the probability of losing the entire bankroll during a game or series of games. It depends on several key factors:

1. Bet size: The bigger the bet, the faster a player can lose his bankroll, especially in cases of losing streaks. High stakes increase the likelihood of quick bankruptcy.
2. Bankroll: The size of a player’s bankroll determines how many losses they can sustain before losing all of their funds. The bigger the player’s bankroll, the lower the risk of going broke, as more funds allow longer losing streaks to be endured.
3. House Edge: A casino’s house edge is the average percentage of bets that the casino expects to win as profit over the long term. The higher the house edge, the more the player loses on average for each bet, which increases the risk of going broke.

How to reduce the risk of losing

1. Bankroll Management: One of the most effective ways to reduce the risk of bankruptcy is effective bankroll management. Players should set a limit for the maximum bet size as a proportion of their bankroll. For example, many professional players recommend betting no more than 1-2% of the total bankroll on one bet, which allows you to withstand longer losing streaks.
2. Understanding Variance: Variance determines the variation in winnings and losses in a game. Players should understand that games with high variance, such as slots or single-number bets in roulette, have higher fluctuations and can quickly lead to losses. In contrast, low variance games such as blackjack or baccarat can provide more stable results with less fluctuation.
3. Choosing games with a low house edge: Choosing games with a low house edge, such as blackjack (with basic strategy) or poker (where players compete against each other and not against the casino), can greatly reduce the risk of going broke.
4. Use of strategies: Using mathematical strategies, such as the Kelly criterion, helps to optimize the size of the bet to minimize risk and maximize potential winnings.

The Law of Large Numbers

The Law of Large Numbers (LLN) is a fundamental principle of probability theory that states that as the number of trials increases, the mean of the results will approach the mathematical expectation. There are two main forms of this law:

1. Weak Law of Large Numbers: For any random variable with finite mathematical expectation, the sample mean will converge to the mathematical expectation with probability that approaches 1 as the number of trials approaches infinity.
2. Strong law of large numbers: The sample mean almost certainly converges to the mathematical expectation as the number of trials approaches infinity.

In the context of gambling, such as in a casino, the law of large numbers ensures that an establishment’s advantage becomes more apparent over time. The house edge is the mathematical advantage the casino has in each game. For example, in roulette, the casino has an edge of approximately 5.26% on the American wheel.
Players may experience short-term gains or losses due to chance. However, over time, as the number of games increases, the average result will approach the mathematical expectation that takes into account the house edge. This means that the casino will always be in the black in the long run.

The Gambler’s Fallacy

Gambler’s fallacy (also known as “player’s error” or “Monte Carlo error”) is the mistaken belief that past events can influence future outcomes in independent events. This means that people believe that if a certain event happened often in the past, it is less likely to happen in the future, or vice versa.

1. Roulette: One of the most common examples of a gambler’s delusion is the belief that after a series of reds on the roulette wheel, the next result “must” be black. In fact, each spin of the wheel is an independent event, and the probability of rolling red or black remains the same.
2. Coin Toss: If a coin comes up heads five times in a row, many believe that it “must” come up tails the next time. However, the probability of an “eagle” or “tail” always remains 50%.

Player delusion can lead to bad decisions and unnecessary losses. Here’s how it can happen:

1. Misbetting: Players may place large bets believing they are “guaranteed” to win after a losing streak. This can lead to significant financial losses.
2. Psychological stress: Constant losses can cause stress and frustration, which can affect a player’s ability to make rational decisions.
3. Gambling Addiction: The belief that “next time will be lucky” can motivate gamblers to continue gambling even when they have already suffered significant losses.