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Designed by Linda Mark Library of Congress Cataloging-in-Publication Data Kucharski, Adam (Mathematician) The perfect bet: how science and math are taking the luck out of gambling / Adam Kucharski. pages cm Includes bibliographical references and index. ISBN 978-0-465-09859-0 (ebook) 1. Games of chance (Mathematics) 2. Gambling. 3. Gambling systems. 4. Probabilities. I. Title. QA271.K83 2015 519.2’7—dc23 2015034255 10 9 8 7 6 5 4 3 2 1 For my parents Luck is probability taken personally. —CHIP DENMAN CONTENTS Introduction Chapter 1: The Three Degrees of Ignorance Chapter 2: A Brute Force Business Chapter 3: From Los Alamos to Monte Carlo Chapter 4: Pundits with PhDs Chapter 5: Rise of the Robots Chapter 6: Life Consists of Bluffing Chapter 7: The Model Opponent Chapter 8: Beyond Card Counting Acknowledgments Notes Index INTRODUCTION IN JUNE 2009, A BRITISH NEWSPAPER TOLD THE STORY OF ELLIOTT Short, a former financial trader who’d made over £20 million betting on horse races. He had a chauffeur-driven Mercedes, kept an office in the exclusive Knightsbridge district of London, and regularly ran up huge bar tabs in the city’s best clubs. According to the article, Short’s winning strategy was simple: always bet against the favorite. Because the highest-rated horse doesn’t always win, it was possible to make a fortune using this approach. Thanks to his system, Short had made huge profits on some of Britain’s best-known races, from £1.5 million at Cheltenham Festival to £3 million at Royal Ascot. There was just one problem: the story wasn’t entirely true. The profitable bets that Short claimed to have made at Cheltenham and Ascot had never been placed. Having persuaded investors to pour hundreds of thousands of pounds into his betting system, he’d spent much of the money on holidays and nights out. Eventually, his investors started asking questions, and Short was arrested. When the case went to trial in April 2013, Short was found guilty of nine counts of fraud and was sentenced to five years in prison. It might seem surprising that so many people were taken in. But there is something seductive about the idea of a perfect betting system. Stories of successful gambling go against the notion that casinos and bookmakers are unbeatable. They imply that there are flaws in games of chance, and that these can be exploited by anyone sharp enough to spot them. Randomness can be reasoned with, and fortune controlled by formulae. The idea is so appealing that, for as long as many games have existed, people have tried to find ways to beat them. Yet the search for the perfect bet has not only influenced gamblers. Throughout history, wagers have transformed our entire understanding of luck. WHEN THE FIRST ROULETTE wheels appeared in Parisian casinos in the eighteenth century, it did not take long for players to conjure up new betting systems. Most of the strategies came with attractive names, and atrocious success rates. One was called “the martingale.” The system had evolved from a tactic used in bar games and was rumored to be foolproof. As its reputation spread, it became incredibly popular among local players. The martingale involved placing bets on black or red. The color didn’t matter; it was the stake that was important. Rather than betting the same amount each time, a player would double up after a loss. When players eventually picked the right color, they would therefore win back all the money lost on earlier bets plus a profit equal to their initial stake. At first glance, the system seemed flawless. But it had one major drawback: sometimes the required bet size would increase far beyond what the gambler, or even casino, could afford. Following the martingale might earn a player a small profit initially, but in the long run solvency would always get in the way of strategy. Although the martingale might have been popular, it was a tactic that no one could afford to carry out successfully. “The martingale is as elusive as the soul,” as writer Alexandre Dumas put it. One of the reasons the strategy lured in so many players—and continues to do so—is that mathematically it appears perfect. Write down the amount you’ve bet and the amount you could win, and you’ll always come out on top. The calculations have a flaw only when they meet reality. On paper, the martingale seems to work fine; in practical terms, it’s hopeless. When it comes to gambling, understanding the theory behind a game can make all the difference. But what if that theory hasn’t been invented yet? During the Renaissance, Gerolamo Cardano was an avid gambler. Having frittered away his inheritance, he decided to make his fortune by betting. For Cardano, this meant measuring how likely random events were. Probability as we know it did not exist in Cardano’s era. There were no laws about chance events, no rules about how likely something was. If someone rolled two sixes while playing dice, it was simply good luck. For many games, nobody knew precisely what a “fair” wager should be. Cardano was one of the first to spot that such games could be analyzed mathematically. He realized that navigating the world of chance meant understanding where its boundaries lay. He would therefore look at the collection of all possible outcomes, and then home in on the ones that were of interest. Although two dice could land in thirty-six different arrangements, there was only one way to get two sixes. He also worked out how to deal with multiple random events, deriving “Cardano’s formula” to calculate the correct