Key Takeaway:
The origins of commercial gambling can be traced back to the mid-1600s when mathematical probability emerged. In 1713, mathematician brothers Johann and Jacob Bernoulli introduced the “Golden Theorem,” later known as the law of large numbers, or long averages. This led to a “probability revolution” in gambling, transforming the industry in Britain and setting the stage for a global change in games of chance. The law of large numbers applies to events governed by probability, such as the “mean of probability.” Despite the complexity of the law, gambling operators continued to innovate, leading to the introduction of “even money” betting in the 1740s. This change not only shaped the future of gambling but also set a precedent for industries where a seemingly small advantage could yield substantial returns over time.
The origins of commercial gambling can be traced back to a time when science was beginning to grasp the mysteries of chance. In the mid-1600s, as mathematical probability emerged, so too did the first regulated gambling houses. By the early 1700s, cities like London and Paris were abuzz with establishments where the thrill of a win—and the odds of a loss—drew eager players. However, many of these games had odds steeply stacked against the players, a setup gamblers took for granted until a new understanding of probability came into play.
In 1713, mathematician brothers Johann and Jacob Bernoulli introduced the “Golden Theorem,” later known as the law of large numbers, or long averages. Surprisingly, though, it took some time before the gambling industry realized the theorem’s full potential. The brothers’ insights showed that a smaller house advantage could yield immense profits over time. But it would take government intervention, regulation, and some creative adaptation for gambling operators to appreciate just how lucrative even a minimal edge could be.
As laws began restricting certain forms of betting, gambling operators started rethinking the math behind their games. This led to a “probability revolution” in gambling, transforming the industry in Britain and setting the stage for a global change in games of chance.
A Game-Changing Theorem and the Potential for Massive Gains
The law of large numbers applies to events governed by probability. When flipping a coin, for instance, there’s a 50% chance it lands on heads or tails. Flip it ten times, and the results might be uneven, but after hundreds or thousands of flips, the tally will naturally trend closer to an even split. This is the “mean of probability.”
In the 18th century, probability theories popularized by mathematician Abraham De Moivre explained how, with enough time, a party with even a slight advantage could ultimately win the bulk of money wagered. Roulette exemplifies this: with 36 numbered slots, 18 red and 18 black, two extra green slots—0 and 00—give the house its edge. Every time the ball lands in one of these green slots, the house collects. Over hundreds of spins, this edge proves reliably profitable.
Imagine 10 players with $100 each, half betting on red and half on black. With an edge that takes effect on average once every 19 spins, the house could take all the players’ money after 200 spins, and with a single zero slot instead of two, the house would claim every dollar after 400 spins. This inevitability, as De Moivre noted, was almost “incredible” given the small advantage.
Reluctance to Apply the Math
Despite De Moivre’s groundbreaking explanations, his insights weren’t immediately adopted by gamblers or operators. Many gamblers found his math too complex, and operators stuck with games that offered clear and long odds, like lotteries, believing these games safer than De Moivre’s unproven theories.
Popular games included the Royal Oak Lottery, played with a 32-sided die. Players could bet on one or more numbers, with winning odds set at 7-to-1 at best. Another was faro, where players could defer winnings to chase larger payouts at escalating odds. Venues like taverns and coffeehouses offered these games, which quickly turned profitable. Although keeping a gaming house was illegal, loopholes and inconsistent enforcement allowed operations to thrive. Public disapproval, however, led to government intervention: the Lottery Act of 1699 and subsequent laws in the 1730s and 1740s targeted games like faro, labeling them as deceptive lotteries due to the unclear odds for players.
Bringing the Law of Averages into Play
The government’s crackdown on games of chance forced gambling operators to test the law of large numbers. By outlawing games played with dice, cards, or numbered devices, authorities sought to curtail these operations. But instead of discouraging gambling, these measures prompted operators to innovate.
Enter the game EO, named after the terms “even” and “odd.” Launched in the 1740s, it was a workaround that cleverly skirted the latest gambling restrictions. EO was played on a wheel with 40 slots, mostly marked “E” or “O,” with two unmarked “bar holes.” As in roulette, if the ball landed in one of these bar holes, the house took all bets. The near-equal odds of EO appealed to players, but the house’s built-in advantage ensured profits. Though some saw EO’s justification as shaky, overburdened law enforcement allowed it to spread.
The Appeal of “Even Money”
Operators soon discovered that games with close odds enticed players to bet more. EO’s popularity inspired a shift in the industry, with casinos on both sides of the English Channel introducing “even money” betting to new and existing games.
In France, biribi was a favorite among gamblers in the 18th century. In this game, players bet on numbers from 1 to 72, and the winning number was drawn randomly. Initially, biribi offered high-risk bets with steep odds, but as the century wore on, operators added even-money options: bets could be placed on whether the winning number would fall within a certain range or on a red or black outcome, laying the groundwork for roulette as we know it today.
In Britain, moral and social pressures eventually led to the Gambling Act of 1845, which banned public and private games of chance for money. However, by then, British players had options beyond their own borders. They could travel to Europe, where casinos were adopting these more mathematically sound games that offered better odds and established gambling as a formidable, profit-making venture.
By marrying probability theory with cunning business strategy, the gambling industry reinvented itself, turning small house edges into large profits. This change not only shaped the future of gambling but also set a precedent for industries where a seemingly small advantage could yield substantial returns over time.
