Successful betting requires both patience and intelligence to derive an advantage from uncertainty.

Source: Qile Club

Adam Kucharski's 'The Perfect Bet' is a story about how to use systematic probability analysis, mathematical models, and science to win. As Kucharski puts it, this is 'the science behind the odds.' The science of betting has been a subject of interest for many years. Throughout history, the act of 'placing bets' has fundamentally transformed humanity’s understanding of luck.

The idea of a perfect betting system is alluring. Stories of making significant profits through betting challenge the perception that casinos and bookmakers are invincible, but these stories imply that there are loopholes in games of probability. If we are smart enough to discover and exploit these loopholes, randomness can be rationally explained, and luck might even be controlled by formulas.

1. A Pipeline for Generating Remarkable Ideas

Kucharski views 'odds' as a science. Interestingly, those who study 'odds' are not all gamblers but include renowned mathematicians and physicists.

As early as the Renaissance, Italian mathematician Gerolamo Cardano developed a formula known as 'Cardano's Formula,' which solves cubic equations. Cardano himself enjoyed gambling and used it to measure the probabilities of random events. In Cardano’s time, what we now know as probability theory had not yet been formulated; there were no laws governing chance events or patterns concerning the likelihood of occurrences. If someone rolled two sixes with dice, it was purely attributed to good fortune. For many gambling games, players did not understand what constituted a 'fair' bet.

Cardano was one of the individuals who discovered that gambling games could be analyzed using mathematical knowledge. He realized that to find the right direction in a world dominated by luck, one must identify its boundaries. Therefore, he would examine all possible outcomes and then focus on those that interested him.

In the following decades, other researchers gradually unraveled the mysteries of probability. Galileo Galilei studied why certain combinations of numbers appeared more frequently than others. Johannes Kepler, while researching planetary motion, wrote a brief article on dice and betting theory.

In 1654, the science of probability flourished thanks to a gambling problem posed by French writer Antoine Gombaud. His dilemma was: Is it easier to roll a six with one die in four throws or to roll double sixes with two dice in 24 throws? He believed the probabilities were the same but couldn't prove it, so he sought advice from mathematician Blaise Pascal.

To solve this problem, Pascal collaborated with the great mathematician Pierre de Fermat. Building on Cardano’s findings regarding randomness, they jointly established the fundamental laws of probability theory. They defined the expected value of a game, which measures the average return from repeated gambling. Their research proved that Gombaud’s assumption was incorrect: it is easier to roll a six with one die in four throws than to roll double sixes with two dice in 24 throws. Thanks to Gombaud's question, an entirely new set of ideas emerged in mathematics.

By the 18th century, Swiss mathematician Daniel Bernoulli approached betting problems by considering 'expected utility' rather than expected monetary gains. He argued that the amount of money one possesses determines the relative value of the same sum of money. For example, a coin is worth more to a poor person than to a wealthy individual. This insight was remarkably profound and laid the foundation for the entire insurance industry based on the concept of utility.

In modern society, betting continues to influence scientific thought. It spans fields such as game theory, statistics, chaos theory, and artificial intelligence. After all, betting serves as a window into the realm of luck, demonstrating how to balance risk and reward, and why people’s valuation of things varies by context. It helps us understand decision-making processes and how to mitigate the effects of chance. Betting encompasses mathematics, psychology, economics, and physics, attracting researchers interested in random or seemingly random events. Often, science and betting form a complete loop: methods initially born out of academic interest have led to attempts to beat the house in practice.

In the late 1940s, physicist Richard Feynman visited Las Vegas. He experimented with various betting strategies to estimate how much he could win or might lose. However, he quickly lost. Feynman met a professional gambler named Nick Dandolos, who always seemed to win. Dandolos told Feynman that he only placed bets when the odds were in his favor. Rather than gambling at the tables, he bet against others around the table—those superstitious about lucky numbers or influenced by biases. Understanding that the casino always had the upper hand, he wagered against inexperienced players. While obvious strategies would lead to losses, Dandolos devised methods to tilt the odds in his favor.

Calculating numbers has never been the hardest part; the real skill lies in transforming them into effective strategies. Betting has consistently given rise to new scientific domains, inspiring fresh insights into luck and decision-making. These insights have significantly impacted areas such as technology and finance. From simple to complex, bold to absurd, betting acts as a pipeline for astonishing ideas. Charlie Munger likened investing to betting. Bettors worldwide challenge the limits of predictability, striving to navigate the boundary between order and chaos. Some explore the mysteries of decision-making and competition, while others observe peculiarities in human behavior and probe the nature of intelligence. By analyzing successful betting strategies, we uncover why betting influences our understanding of luck and how we can harness it to our advantage.

II. Sensitive Dependence on Initial Conditions

Physicist Henri Poincaré was deeply intrigued by roulette. In his view, the outcome of roulette appeared random because we lacked an understanding of its underlying principles. He proposed that problems could be classified based on levels of ignorance. If we know the precise initial state of an object—such as its position and velocity—and the physical laws it follows, solving the problem becomes a textbook exercise. Poincaré termed this “first-order ignorance”: possessing all necessary information but requiring straightforward calculations. “Second-order ignorance” refers to knowing the physical laws but lacking knowledge of the exact initial state or being unable to measure it accurately. In such cases, either measurement techniques must improve, or predictions must remain confined to narrow parameters. “Third-order ignorance” represents the broadest level of ignorance: not knowing the initial state or the governing physical laws. When these laws are too complex to decipher, individuals find themselves in third-order ignorance.

Roulette is no exception. The trajectory of the ball depends on numerous factors, making them difficult to discern merely by observing the spinning wheel. According to Poincaré, it is unnecessary to know why the ball lands in a specific position; instead, one needs to observe many spins and analyze the outcomes. This approach was precisely what artificial intelligence scientist Albert Hibbs and pathologist Roy Walford undertook in 1947. Casinos rely on the assumption that each number on the roulette wheel has an equal chance of appearing, but like any machine, roulette tables may develop defects or wear over time. They sought tables where the distribution of numbers became uneven.

Poincaré argued that minor differences in the ball's initial state could lead to significant variations in outcomes—variations too large to ignore but too subtle to notice. Consequently, we perceive the results as random. This phenomenon is known as “sensitive dependence on initial conditions,” implying that detailed measurements of a process, whether roulette spins or tropical storms, may overlook minor events with major consequences. Meteorologist Edward Lorenz once asked in a lecture, “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” Unbeknownst to him, Poincaré had already outlined the concept of the “butterfly effect” seventy years earlier.

Lorenz’s research eventually evolved into “chaos theory,” primarily used for prediction. His goal was to improve weather forecasting and develop methods to predict weather patterns over longer periods. Conversely, Poincaré was interested in the opposite question: how long does it take for a process to become random? Does the path of the roulette ball truly become random?

While roulette inspired Poincaré, his breakthrough came from studying larger-scale trajectories. Poincaré demonstrated that “sensitive dependence on initial conditions” also applies to asteroid orbits. To him, the zodiac and roulette table were merely two examples of the same theory. He believed that after enough spins, the ball’s final position would be entirely random. He also noted that persistently betting on certain choices would reveal randomness earlier than others.

Successful roulette strategies rest on casinos’ confidence that the outcomes of roulette spins cannot be predicted. The evolution of these strategies reflects the development of probability science in the 20th century. Early attempts to beat roulette focused on escaping what Poincaré called third-order ignorance—the state of being completely unaware of physical processes. Karl Pearson’s research was purely statistical, aiming to uncover patterns in data. Later efforts to profit from gambling adopted different approaches, attempting to overcome second-order ignorance: the sensitive dependence of outcomes on the initial conditions of the roulette wheel and ball.

For Poincaré, roulette was merely a means to illustrate his ideas: simple physical processes could gradually descend into randomness. This notion became a critical component of chaos theory and contributed to the emergence of an entirely new academic discipline in the 1970s.

3. Sensitive Dependence on Initial Conditions

Among these 'new' academic fields emerged John von Neumann's 'game theory,' John Nash's 'Nash equilibrium,' Ronald Fisher's 'extreme value theory,' Edward Thorp's 'winning strategy for blackjack,' John Kelly's 'Kelly criterion,' and more.

However, betting appears to differ significantly from other investment categories. During the 2008 financial crisis, many asset prices plummeted. Investors often attempt to construct portfolios resilient to such shocks by holding shares in companies across various industries. Yet when markets fail, this diversity proves insufficient to mitigate risk. According to Tobias Preis, a researcher of complex systems at the University of Warwick in the UK, stocks tend to behave similarly during tough times in financial markets. Preis analyzed stock prices in the Dow Jones Index from 1939 to 2010 and found that under market stress, stock prices generally decline. Thus, the risk-diversification effect intended to protect a portfolio disappears during periods of market losses.

This issue is not confined to equities. Just before the onset of the 2008 financial crisis, an increasing number of investors began trading 'collateralized debt obligations' (CDOs). These financial products bundle unpaid loans, such as mortgages, allowing investors to profit by assuming part of the borrowers' risk. Although the likelihood of any single borrower defaulting was high, investors believed that simultaneous defaults by all borrowers were highly improbable. However, this assumption proved incorrect. Following the financial crisis, when one property lost value, others followed suit.

The optimal strategy focuses not on 'how to win the most' but on 'how to lose the least.' Over time, both the house and gamblers gradually master the most well-known strategies, making it difficult to profit from them. People tend to learn strategies through 'experience-weighted attraction,' favoring actions that have succeeded in the past. The most successful individuals are often those who study what others overlook. Consequently, some opt for 'counterintuitive' strategies rather than those that may actually be superior from a game-theoretic perspective.

Research has shown that as the number of players increases, disorderly decision-making becomes more prevalent. When games are highly complex, player choices become nearly unpredictable. In the early 1960s, mathematician Benoit Mandelbrot observed financial markets and noted that turbulent periods in the stock market often cluster together. He wrote that large changes tend to follow large changes, while small changes tend to follow small changes. The emergence of 'volatility clustering' captured economists' interest.

Edward Thorp discovered a significant loophole in the card game blackjack, which led to his best-selling book 'Beat the Dealer.' However, the debate over whether victory depends on luck or skill spread to other games. This debate even determined the fate of the once highly profitable U.S. poker industry. In 2011, U.S. authorities shut down several major poker websites, bringing an end to the nationwide 'poker boom.'

Economist Randall Heeb believes poker is a game of skill. He pointed out that top-ranked players consistently win except for a few off days, while less skilled players lose heavily over the course of a year. The fact that some individuals make a living playing poker is undoubtedly proof that the game requires skill. Successful poker players win partly because they can control the situation.

Another economist, David Derosa, disagrees with Heeb’s view on poker. He simulated what would happen if 1,000 people flipped coins 10,000 times using a computer. The results were strikingly similar to Heeb’s findings: a small group of people consistently won, while the rest lost badly. This does not imply that coin flipping involves skill; it simply suggests that rare events are bound to occur when observing a sufficiently large sample size, akin to the 'infinite monkeys' scenario. Therefore, the central question remains: how long must we wait for skill to outweigh luck?

The notion of 'infinite monkeys' originates from mathematician Émile Borel. Borel once provided a classic example: a monkey randomly hitting the keys of a typewriter happens to produce the complete works of Shakespeare. He wrote, 'Although the likelihood of such events cannot be reasonably argued, their probability is so minuscule that any rational person would unhesitatingly consider them impossible.'

4. Equipped with academic knowledge and new technologies

Although roulette has long been regarded as the epitome of randomness, it was first challenged by statistics and then by physics. Other games have also succumbed to science. Poker players utilize game theory, while betting syndicates have transformed sports wagering into an investment. According to Stanislaw Ulam, who researched hydrogen bombs at Los Alamos, the presence of skill in such games is not always obvious. He stated, 'There is such a thing as habitual luck; people tend to believe that someone with exceptionally good fortune in card games might possess hidden talents, which include skill.' Ulam believed this principle also applies to scientific research. Some scientists encounter luck so frequently that it is hard not to suspect an element of inherent talent.

Completely eliminating luck is impossible, but experience shows that it can often be partially replaced by skill. Therefore, seemingly random processes are often not truly random. In chess, there is no inherent randomness. If two players make identical moves each time, the outcome will always be the same. Nevertheless, luck still plays a role because the optimal strategy remains unknown, and a series of random moves could potentially defeat even the best player.

However, when making decisions, our perception of luck is sometimes biased. If the outcome is favorable, we attribute it to skill; if it fails, we blame bad luck. Our perception of skill can also be distorted by external information sources. The media loves stories about entrepreneurs who capitalize on trends to become wealthy or celebrities who suddenly gain widespread fame. We often hear tales of novice writers penning bestsellers or brands achieving overnight success.

Statisticians Mark Roulston and David Hand pointed out that randomness in popularity also affects the rankings of investment funds. 'Suppose fund managers arbitrarily select a group of funds without applying any technical expertise, and some achieve substantial returns purely by chance. These funds will attract investors, while poorly performing ones will close down, disappearing from public view. Looking at the results of surviving funds, one might conclude that they generally involve some degree of skill.'

The dividing line between luck and skill, as well as between betting and investing, is rarely as clear-cut as we imagine. To distinguish between luck and skill in a given context, we must first find a way to measure them. However, sometimes outcomes are highly sensitive to minor changes, where seemingly inconsequential decisions entirely alter the result. Individual events can have dramatic effects, especially in activities like soccer and ice hockey, where goals are rare. In these sports, such events might include a decisive risky pass or a hockey shot that hits the goalpost.

Edward Thorp made a prediction on the final page of his book *Beat the Dealer*: over the next decade, we would witness entirely new methods attempting to tame luck. 'Most of these methods may currently be unimaginable, and their emergence is exciting.' Indeed, the science of betting evolved afterward. It pioneered new fields of study far beyond the physical tables and plastic chips of Las Vegas.

We have seen how roulette helped Henri Poincaré refine early ideas of chaos theory and assisted Karl Pearson in testing his novel statistical techniques. We have also observed how Stanislaw Ulam's card games contributed to the development of the Monte Carlo method, now applied in everything from 3D computer graphics to disease outbreak analysis. Additionally, we saw how game theory emerged from John von Neumann's analysis of poker. As demonstrated, nearly every game can be beaten, but profits rarely come from lucky numbers or foolproof systems. Successful betting requires patience and intelligence. It demands creators who ignore dogma and follow their curiosity.

Probability theory, in particular, stands as one of humanity's most valuable analytical tools, granting us the ability to assess the likelihood of events and evaluate the reliability of information. Consequently, it has become a core component of modern scientific research, ranging from DNA sequencing to particle physics. It is remarkable that a science originating from reflections on games of chance has evolved into one of the most crucial elements of human knowledge. In this field, superstitious thinking has waned, replaced by rigor and research. As Bill Benter, who amassed wealth through blackjack and horse racing, put it, it was not the street-smart gamblers of Las Vegas who devised a system. Success came because outsiders armed with academic knowledge and new technologies entered the scene, illuminating what was once a shadowy domain.

Editor/Jayden