Understanding the link between probability theorem and casinos

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The law of large numbers (LLN) first proved by brothers Johann and Jacob Bernoulli is a fundamental theorem in probability and statistics that states that as the number of trials of a random process increases, the average of the results obtained from those trials will converge to the expected value. This law has significant implications for various fields, including gambling and casino operations.

Law of Large Numbers

Types of LLN

1. Weak Law of Large Numbers (WLLN): This states that for any given positive number, the probability that the average of the results will be close to the expected value approaches 1 as the number of trials goes to infinity.

2. Strong Law of Large Numbers (SLLN): This asserts that the average of the results almost surely converges to the expected value as the number of trials goes to infinity.

Implications of LLN for Casinos

Casinos heavily rely on the LLN to ensure their profitability over the long term. Here’s how:

1. House Edge: Each casino game is designed with a mathematical advantage for the house. For example, in roulette, the presence of the green 0 (and 00 in American roulette) ensures that the expected value is in favor of the casino.

2. Large Volume of Bets: By encouraging a high volume of bets and frequent play, casinos can ensure that their theoretical house edge translates into actual profits. Over many games and bets, the average outcomes will converge to the house edge due to the LLN.

3. Variance Management: In the short term, there can be significant variance, and players might win large sums. However, over a large number of bets, the variance evens out, and the casino’s edge prevails.

4. Comping High Rollers: Casinos often offer incentives to high rollers to keep them playing, knowing that the more they play, the more likely the LLN will work in the casino’s favor.

Historical Context of LLN and Casinos

The connection between gambling and probability theory dates back centuries:

1. Gambling Origins: Gambling is one of the oldest activities, with evidence of it in ancient civilizations, including India,  Mesopotamia, China, and Greece.

2. Probability Theory Development: The formal study of probability began in the 17th century, driven by gamblers’ interest in understanding their chances. Blaise Pascal and Pierre de Fermat’s correspondence laid the foundation for probability theory.

3. Casinos’ Evolution: The first known casino, the Ridotto, opened in Venice in 1638. Modern casinos have evolved from these early establishments, incorporating sophisticated mathematical models to optimize profits as theories related probabilities began to be proved during this period.

4. Mathematics and Casinos: Mathematicians like Gerolamo Cardano, who wrote about probability in the context of gambling, significantly influenced the field. Later, Jacob Bernoulli formally introduced the LLN in his work “Ars Conjectandi” in 1713.

Modern Casino Strategies based on LLN

Casinos today use a variety of strategies, supported by the LLN, to maintain their edge:

1. Game Design: Ensuring that all games have a built-in house edge.

2. Technology: Using surveillance and analytics to monitor patterns and ensure fair play.

3. Regulations: Adhering to regulatory standards that guarantee the integrity of games while maintaining profitability.


The law of large numbers or LLN is a key principle that underpins the operation of casinos. By ensuring that over a large number of bets, the outcomes will converge to the expected probabilities, casinos can consistently secure profits. The historical evolution of casinos and their reliance on mathematical principles illustrate the deep connection between gambling and probability theory.