Part III: Business & Regulation · Gambling Mathematics & Probability

13. Law of Large Numbers

Foundational Probability

The Law of Large Numbers is a fundamental theorem of probability stating that as the number of independent trials increases, the observed average of outcomes will converge toward the expected value. This principle is the mathematical foundation of casino profitability — while individual players may win in the short term, the casino’s mathematical edge guarantees profit across millions of wagers. The law applies to all games of independent chance including roulette, craps, and slot machines. Casino executives and surveillance professionals should understand that short-term deviations from expected results (both positive and negative) are normal, but over sufficient volume, results will always converge to the theoretical house edge.

In practice

This law does NOT mean that past results affect future outcomes (the Gambler’s Fallacy). It states that the proportion of outcomes converges to expected probability, not that individual outcomes “balance out.” The convergence rate is proportional to the square root of the number of trials.

One entry from the Casino Industry Glossary — 1,157 terms written for surveillance, compliance and operations professionals rather than for players. Definitions describe industry usage; where a term carries a regulatory meaning, verify against the instrument that governs your jurisdiction.