Gambling has helped make the modern world. Mathematician Adam Kucharski explains how casinos and card games inspired many ideas that contributed to science.

1. Dice and the Birth of a New Science

In the 16th century, there was no way to quantify luck. If someone got two sixes during a dice game, people thought it was just luck. Gerolamo Cardano, an Italian physician with a lifelong passion for gambling, thought differently. He decided to study games from a mathematical point of view and wrote a game manual that outlines how to navigate in the "variable space" of possible events. For example, two dice can fall in 36 different ways, but only one way produces two sixes.

From this research the concept of the theory of probability was revived. This means we can calculate the likelihood of an event and clearly determine how lucky or unlucky we were. With his new methods, Cardano gained a decisive advantage in the gaming halls, and mathematics was given a completely new field of study.

2. The Dots Problem

Suppose you flip a coin with a friend and the first person to win six flips gets £ 100. How do you split the money if one of you is leading 5-3? In 1654, the French aristocrat Antoine Gombeau asked the mathematicians Pierre de Fermat and Blaise Pascal to help him solve a "point problem" like this one.

In addressing the issue, Fermat and Pascal developed the concept of “expected value”. It determined the average probability of each side winning if the event repeats itself several times before the end of the game. Now this concept is one of the key elements of economics and finance: by calculating the expected amount of investment, we can determine how much of it will fall on each of the participants.

In the case of coin tosses, your friend (who is 5-3 behind you) must hit three winning tosses in a row to win. His chances of doing it are 1: 8, your chances are 7: 8 on average. Thus, the money should be divided in a ratio of 7: 1, i.e. £ 87.50 to £ 12.50.

3. Roulette and Statistics

During the 1890s, the Le Monaco newspaper regularly published the results of roulette spins in Monte Carlo casinos. At the time, this was a godsend for mathematician Karl Pearson. He studied the probabilities of random events and looked for suitable data to test his method. Unfortunately, it turned out that the results of the roulette spins were not entirely random as he had hoped. "If the Monte Carlo roulette wheel has existed since the era of the earth’s geological formation," Pearson noted after examining the data, "we could still not expect a repeat of these two weeks."

Pearson’s methods, honed by the research of roulette, are now an important part of science. From medical testing to experiments at CERN (European Organization for Nuclear Research), researchers test theories by calculating the likelihood of getting the desired result in the form of a coincidence, obtained solely through luck. This allows them to establish whether there is sufficient evidence to support their hypothesis, or whether these results are nothing more than a coincidence. As for the unusual data of roulette in Monte Carlo, which did not fit into Peirce’s theory, the explanation for this phenomenon turned out to be very simple. The fact is that instead of recording the results of the spins, Le Monaco’s lazy journalists decided that it would be easier to just come up with numbers.

4. St. Petersburg Lottery

Let’s say we are playing the following game. I flip the coin several times until it comes up heads. If it comes up heads on the first throw, I pay you £ 2. If it comes up on the second throw for the first time, I give you £ 4; if on the third, I pay £ 8 and so on, doubling the bet each time. How much are you willing to pay me to play this game?

Known as the St. Petersburg Lottery, this game stunned 18th century mathematicians because the perceived value of the game (that is, the average of all payouts if played a large number of times) was enormous. However, only a few people were willing to pay more than a few pounds to play. In 1738, mathematician Daniel Bernoulli solved the puzzle by introducing the concept of "utility." The less money a person has, the less he will want to risk, having a small chance of getting a huge win in a bet. This concept is now central to the economy, and in fact underlies the entire insurance industry. Most of us would prefer to make small, recurring payments to avoid high potential costs, even if we end up paying more.

5. Roulette and Chaos Theory

In 1908, the mathematician Henri Poincaré published Science and Method, in which he reflected on our prediction skills. He noted that games such as roulette seem to be random because small differences in the initial speed of the ball, which are very difficult to measure, can greatly affect the ball’s landing point. In the second half of the 20th century, this "sensitive dependence on initial conditions" will become one of the fundamental concepts of "chaos theory". The goal was to explore the predictability limits of physical and biological systems.

When chaos theory developed into the field of science, the connection with roulette remained. The pioneers of chaos theory in the 1970s were physicists J. Doyne Farmer and Robert Shaw, who during their student days secretly used hidden casino computers to measure the speed of a roulette ball and, using the data, successfully predict the results.

6. Solitaire and the Power of Simulation

Computers play a key role in the theory of probability. One of the main events happened in the 1940s thanks to a mathematician named Stanislav Ulam. Unlike many of his peers, he was not one of those people who like to make lengthy calculations. He was playing Canfield, a form of solitaire that originated in casinos, and wondered what the probability of cards being hit in the optimal sequence to win was. Instead of trying to figure out all the options, he realized that it would be easier to just lay out the cards a few times and see what happens.

In 1947, Ulam and his colleague John von Neumann applied the new Monte Carlo method to study nuclear chain reactions at Los Alamos National Laboratory in New Mexico. Through repeated computer simulations, they were able to solve a problem that was too complex to be solved using traditional mathematics. Since then, the Monte Carlo method has become an important part of other industries, from computer graphics to outbreak analysis.

7. Poker and Game Theory

John von Neumann was outstanding at many things, especially poker. To determine the most effective strategies, he decided to analyze the game in terms of mathematics. While it was probable that it was possible to determine which cards were dealt, solving this one problem was not a sufficient condition for winning: he also had to anticipate what his opponent might do.

Von Neumann’s analysis of the game of poker and baccarat led to the emergence of the field of "game theory", which deals with mathematical decision-making strategies between different players. Among those who grew up with von Neumann’s ideas was John Nash, whose story is told in the movie "A Beautiful Mind". Since then, game theory has found its way into economics, artificial intelligence, and even evolutionary biology. It may not be all that surprising that ideas from the field of gambling have spread to many areas. As von Neumann once remarked, "real life is bluffing."

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