How did casinos make money from the mathematical advantage in their games?

Casinos generate profit by maintaining a mathematical advantage, often called the "vigorish" or house edge, which ensures they win in the long run. This advantage is built into the structure of games through several methods:12

• Payout Discrepancies: Casinos often pay winners less than the "fair odds" of an event occurring. For example, in craps, the house may return significantly less than the true probability of a winning outcome would dictate.2
• Game Rules: Rules are specifically designed to favor the house. In roulette, the inclusion of 0 and 00 on American wheels creates an advantage of approximately 5.26 percent for the bank. Similarly, in blackjack, casinos use multiple decks and reshuffling procedures to minimize the impact of card counting, a strategy players use to try to overcome the house edge.123
• Expected Value: Casinos use expected value to set the cost of play. By ensuring the cost to play is higher than the statistical expected return to the player, the casino guarantees a profit over a large number of trials.4

While individual outcomes are governed by chance, the law of large numbers ensures that the house's mathematical edge becomes increasingly reliable as the number of games played increases.2