What's Luck Got to Do with It?: The History, Mathematics, and Psychology of the Gambler's Illusionby Joseph Mazur
Why do so many gamblers risk it all when they know the odds of winning are against them? Why do they believe dice are "hot" in a winning streak? Why do we expect heads on a coin toss after several flips have turned up tails? What's Luck Got to Do with It? takes a lively and eye-opening look at the mathematics, history, and psychology of gambling to reveal/i>
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Why do so many gamblers risk it all when they know the odds of winning are against them? Why do they believe dice are "hot" in a winning streak? Why do we expect heads on a coin toss after several flips have turned up tails? What's Luck Got to Do with It? takes a lively and eye-opening look at the mathematics, history, and psychology of gambling to reveal the most widely held misconceptions about luck. It exposes the hazards of feeling lucky, and uses the mathematics of predictable outcomes to show when our chances of winning are actually good.
Mathematician Joseph Mazur traces the history of gambling from the earliest known archaeological evidence of dice playing among Neolithic peoples to the first systematic mathematical studies of games of chance during the Renaissance, from government-administered lotteries to the glittering seductions of grand casinos, and on to the global economic crisis brought on by financiers' trillion-dollar bets. Using plenty of engaging anecdotes, Mazur explains the mathematics behind gambling--including the laws of probability, statistics, betting against expectations, and the law of large numbers--and describes the psychological and emotional factors that entice people to put their faith in winning that ever-elusive jackpot despite its mathematical improbability.
As entertaining as it is informative, What's Luck Got to Do with It? demonstrates the pervasive nature of our belief in luck and the deceptive psychology of winning and losing.
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"In What's Luck Got to Do With It?, mathematician Joseph Mazur explores these misconceptions, taking the reader on an entertaining and accessible tour of the history of gambling, the way mathematicians quantify luck and the psychology that keeps gamblers returning to the table. A book worth taking a chance on."New Scientist
"Doubtless aimed at the interested gambler, the frequent cultural references, anecdotes and intervention of psychology nevertheless make the book appealing reading."Times Higher Education
"Both an analysis of the idea of luck, the gambling impulse, and a history of it, stretching back to Neolithic times, the Renaissance (Francis Drake and Ben Johnson often played hazardan early form of dice) up to the age of one-arm bandits."Steven Carroll, The Age
"Because Mazur's not judgmental about luck and gambling, but is analytical, the book is a winner. It's not just a mathematician telling us that we'll never hit a million-dollar jackpotit's a mathematician looking at why we continue to hope to hit that jackpot. This book should be required reading for anyone in the casino business, and anyone who spends more than a fraction of their disposable income on gambling should find it informative, if nothing else. It's a reasoned, but also passionate, search for the meaning of luck that may change the way you look at a pair of diceor your mortgage."dieiscast.com
"What's Luck Got to Do with It? is an entertaining and informative history of gambling beginning with the Ice Age. . . . Anyone who has an interest in probability will enjoy Mazur's ideas and insights."Mathematics Teacher
"Readers will find many an unexpected treat in Mazur's exploration of luck, or, as Mazur might say, the likelihood of long runs of desired outcomes within the purview of the law of large numbers."Andrew James Simpson,Mathematical Reviews Clippings
"Mazur's book is appealing to virtually anyone with an interest in the human psyche. It should certainly be given out to anyone arriving for work on their first day on Wall Street. Perhaps it would help to avoid a few more disasters."Sam Marsden, Jackpot Gaming Limited
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What's Luck Got to Do with It?The History, Mathematics, and Psychology behind the Gambler's Illusion
By JOSEPH MAZUR
PRINCETON UNIVERSITY PRESSCopyright © 2010 Joseph Mazur
All right reserved.
Chapter OnePits, Pebbles, and Bones
Rolling to Discover Fate
Gaming is a principle inherent in human nature. -Edmund Burke, British House of Commons Record, February 11, 1780
Imagine life in the last Ice Age. Those Neanderthals, with their orangutan jaws and beetle brows, burbling some mono-vowel language, sharpening spears in preparation for a hunt of hungry scimitar-toothed black tigers, reflexively gambling every day against the impending extinction of their race. Ground tremors, as common as cloudy days, triggered by great weights of melting ice continually relaxing gargantuan pressures of the earth's indomitable crust; ordeals of menacing elements, snow and freezing temperatures; hunger, pain, and weakness from the bruises of long, fierce hunts; and most worrisome of all, the daily threats of nearby ravenous beasts stealthily looking for supper. Humans have been gambling ever since that unfathomable island of time, when herds of pachyderms and hump-shouldered mammoths freely wandered over the frozen lakes of the Neander Valley.
Our extinct fellow proto-humans looked brutally intimidating and menacing with their powerful muscles, fleshy fingers, and massive limbs, but they were innocently carrying fierce looks for passive self-protection. They cared for their young, who played with discarded dry bones by kicking, banging, and tossing them here and there in amusement or quite possibly in a shilly-shally appreciation of primordial sport. How natural it is for a child to create games by hurling things. We can readily imagine a Neanderthal child uttering, "My bone for your two that I can throw mine further," even in a primitive language of consonants mixed with ah as its only vowel.
We can picture the adults, in their rare, brief leisure time, wagering on who can throw the farthest spear or on who can down the nearest rhino. They may have tossed bones in games the way kids now toss marbles, laugh when something is funny, or cry when injured. Neanderthals smiled to express joy, frowned to convey displeasure, embraced in camaraderie, and gambled every day with their own lives in decisions of whether or not to go out on a hunt or to wander far from their recognized comfortable safety zones where the fire was warm.
Risks are the gambles, the games, the balance of expectation and fate. And luck rarely comes without risking the possibility of loss, injury, trouble, vulnerability, ruin, or damage in a universe of opposing chances. We also know (from bone sample evidence) that our Neanderthal friends were subjected to a high rate of injury during their lifetimes, most likely from close-range hunting of fast and ferocious sabertooths, whose sword-like canines could effortlessly pierce and slash the skin of even the toughest males. And if the cats-those felidae with the courage to raze mighty mammoths-didn't occasionally slash those hardy men to death, then the bruises those cats inflicted surely disabled them. That was the true gamble-to eat or be eaten.
Gambling is about odds, the chances of things going one way or another. Will the team bring down the mammoth, or will a tusk lethally impale someone? So we humans are programmed to gamble. It's not only about the team. We take risks, leave our houses and explore the uncertain boundaries of secure and reliable neighborhoods, all as part of the animal nature of survival. The urge to take risks is just one of the hardwired universals of being human, along with smiles, frowns, cries, and laughs.
When we come closer to our own time-not much closer, still in the late Pleistocene era, 10,000 to 40,000 years ago-we find our early dark-skinned Cro-Magnon ancestors in the Danube Valley painting on the walls of caves with sticks of charcoal, carving moon calendars or in performance of some ritual event to applaud the supernatural owner of the wildlife they hunt or to thank the shamans. They, too, performed risks and had to ask: Is it safe to paint today? Should we leave home and take care not to be mauled by the nearest beast? Or should we stay protected by warm fire and eat the spoils of yesterday's kill? It was all a gamble at a time when humans were skillfully tuning maneuvers of feet, arms, and wrists to influence and direct flights of their sharp weapons.
Their tools, those spears, arrows, flints, and fires, gave them hunting advantages their brawny Neanderthal neighbors never had: the opportunity to hunt from safe distances. And with them came prophesizing games and innocent gambling. Innocent, because players were not necessarily wagering their fine spears or furs nor-what should have been quite reasonable-staking their best pickings, but rather banking on the moods of randomness for providential guidance and help from the phantoms of predictability in forecasting decisions. A shaman might roll a pair of, say, sheep astragals (anklebones) to determine if the tribe should go out on a hunt the next day. Die-like objects such as filed and sanded astragals have been found in abundance at archaeological sites almost everywhere from central France to as far east as the Punjab. What they were used for is anyone's guess, but more likely than not they brought some form of entertainment or a means of communication with the gods. Someone would ask a question, and, depending on how the astragals fell from the shaman's hand, there would be an answer. One answer was accepted when wide sides faced upward, another when the narrow. Surely these bones were biased; however, it did not take long for our clever ancestors to find a way of evening the odds by rasping the sides of an astragal and smoothening out six faces of a stone or piece of wood for fairer outcomes in inventing the die. Certainly, sticks and odd-shaped stones must have been used for playing against chance. Fruit pits, pebbles, shells, teeth, seeds, and acorns must have given hints of rolling to discover fate.
When we come to the more modern hunting and gathering societies of 10,000 years ago, we see that gambling advances with better impressions of randomness. We find dice being made from carefully carved pieces of wood, stone, and ivory. It's more than just the archaeological findings; now we have the literature, albeit through oral tradition, to back it up. Homer's Iliad tells us that gambling and chance have their roots in the beginning of time when Zeus, Poseidon, and Hades drew lots for shares in the universe.
Lot is the etymological root in the words allotment and lottery. It is also something that happens to a person when the lot has fallen-it was his lot. The casting of lots would have meant any decision-making procedure or mechanism, the flip of a coin, the roll of a die, the pick of a straw. The lot itself might have been an object such as a piece of wood, a pebble, a die, a coin, or a straw that could be used as one of the counters in determining answers to vital questions by the position in which it comes to rest after being tossed or picked. But if the lot is to be fair, it must be far more unbiased than an astragal, which surely does not have equal chances of falling on any of its four sides.
Drawing lots was thought to be the fair way to settle a choice that could not be established by reason. And since every lot banks on the whim of Fortune, or on the very misunderstood impulse of randomness, it might be said that every unreasoned choice is a gamble. Indeed, children of all ancient cultures must have muttered some variant of eenie meenie mynie mo among friends making a choice. Still, for the adults, it was more likely thought of as a means of communication with some supernatural spirit. Getting the short end of the stick might have been the random pick of the draw, but it could also have simply been the will of God. The Mishnah (the section of the Talmud connected to oral laws) says that to draw lots one must have an urn of tablets marked to describe a fate and that those tablets must be alike in size and shape so that any pick is as likely as any other. The Bible says that to atone for the sins of his house, Aaron was to cast lots to decide which of two goats would be sacrificed and which would be sent back into the wilderness.
In Exodus there is a vague description of the breastplate of judgment, a part of Aaron's priestly garment. Aaron was to wear it on entering a holy place to seek God's judgment on difficult questions affecting the welfare of Israel. The breastplate of judgment refers to a vestment of embroidered fabric set with twelve precious stones representing the twelve tribes of Israel and worn by any of the high priests seeking God's guidance on matters concerning the welfare of the community. According to instructions outlined in Exodus, the breastplate must contain the so-called Urim and Thummim, which literally translate as "the Lights and the Perfections." According to some modern commentators, these were two sacred lots, chance instruments such as coins or dice, used for the purpose of determining the will of God on questions of national importance. It may have been that the priest would cast the lots but also understood that while the lot is cast, God manipulates the lot to determine the outcome-"The lot is cast into the lap; But the whole disposing thereof is of the Lord" (Prov. 16:33).
We take "the Lights and the Perfections" here to mean the perfect determination of the truth by means of unbiased casting of lots-the perfect throw of perfect dice for the fairest possible decision.
Fairness is, unfortunately, seldom a functioning human trait, but when it comes to decision-making, inequality is inherently recognizable. The child who must share a piece of cake after being given the opportunity of dividing and cutting it to permit others to choose pieces will try to divide and cut with fairest precision. Humans can recognize overt inequities. So it shouldn't have taken many rolls of astragals to upset early gamblers and cause them to think of a fairer, more random way to cast lots. Though the typical astragal is shaped somewhat like an elongated cube, it has only four sides to fall on; its two end faces are so uneven and knuckly it would be highly unlikely for it to remain standing on one face. Yet, astragals were used for centuries before real cubical dice took over, sometime long before the first millennium BCE, when cubes that could (more or less) fall fairly on one of six faces were introduced. Since then, dice variations have been used in every part of the world from America to Japan, from Sweden to Africa. Recent (2004) archaeological digs in the Bronze Age city of Shahr-I Shokhta (literally, the Burnt City) in southeast Iran unearthed a five-thousand-year-old backgammon set made of ebony with cubical dice.
Dice playing enters stories such as the ancient Indian Sanskrit epic poem Mahabharata, the fifth century BCE tale of Cyrus the Great, and the story of Isis and Osiris in which we learn of the Egyptian game of tau (akin to English draughts or checkers, which goes back to at least 1600 BCE). Older still is the Royal Game of Ur, going back to before 2500 BCE. Two players would race their pieces from one end of the board to the other according to moves controlled by specific landings of a die, a prototype of backgammon. The die would have been either a four-sided stick or tetrahedron.
Modern dice, numbered as ours with opposite sides summing to 7, have been found at archaeological sites in Thebes and elsewhere in Egypt. And we have evidence that the Egyptians played a game called atep-a game still played almost everywhere in the world and one I recall playing as a kid when we had to either choose sides or choose who would go first in a game. We'd call out odds or evens and then extend either one or two fingers on the split second after calling out 1-2-3-shoot. For such a game there are no physical lots but rather mental choices (as if flipping two coins at once) to make fair decisions.
When we come to the Romans we find gambling rampant, though we also find evidence for the first laws against gambling to dampen uncontrolled behavior associated with gaming. It was a time steeped in sexual marathons and drinking sessions mixed with gambling by dice, cards, and quail fights. In nineteenth-century archaeological excavations of Rome, hundreds of gaming tables were found. The tables were typically designed as tabula lusoria (table of play) in the form of three horizontal lines, each containing twelve signs with words arranged to make a sentence with thirty-six letters. The taverns patronized by gamblers used such poetic forms in their signs to warn against fighting over games, no doubt swayed by the thirty-six (6 x 6) distinct possible results of throwing two dice.
LEVATE LVDERE NESCIS DALVSO RILOCV RECEDE
These six terms with thirty-six letters are abbreviations of words that form a sort of haiku rune that unravels to this rough translation: Rise! If you don't know the game make room for better players! Such tabula lusoria would signify good or bad luck, warn of the skill needed to play well, or warn of the risks of gambling. Others were unguarded invitations to gamble.
We know from Plutarch that Marc Anthony was a consummate gambler; that Augustus was an ardent dice player; and that Nero played some variant of craps. Claudius had a special carriage designed for playing dice; he even wrote a book on dice playing. And Caligula, after losing all his money at an ancient variant of craps, ran into the street, confiscated money from two Roman guards, and returned to his game.
Dice have been found all along trails used by the crusaders. Throughout the Middle Ages, from northern Europe to Brindisi at the heel of Italy, crusading armies played dice games at taverns along their way.
Late in the thirteenth century, Alfonso X, king of León and Castile, commissioned the writing of a book of games. Known as Libro de Los Juegos (Book of games), it contains descriptions and illuminated illustrations of all sorts of games from chess to backgammon, including dice and tables.
The story told in Alfonso's book of games is that there was once a king who would often consult his three wise men over the nature of things, and on one particular occasion the debate came to the question of gaming and of the advantages of luck and brains. One wise man said that brains were more valuable than luck because thinking gives order to life and even if he lost, he would not be to blame because he used reason. Another said that luck was more valuable than brains because win or lose, his brains could not avoid his destiny. And the third said it would be better to have both-to use the brain to his reasoning advantage and to use luck to protect him from any potential harm.
Alfonso was on to something that was to become the core understanding of all professional gambling from cards to hedge funds. The balance of luck and reasoning could be interpreted through a rational measure of how favorable the outcome might be. Though Alfonso had no conception of risk management and certainly no perception of positive expectancy (the mathematical tool that modern professionals use to quantify a future event), he did understand that the blends of luck and skill behind gambling games fall into a wide spectrum.
Alfonso's book of games tells us that dice are perfect cubes, made of wood, stone, bone, or, best of all, metal-as perfect as thirteenth-century craftsmen could manufacture-otherwise they would roll more often on one side than any other and would be a trick of luck. The spots are marked just as they would be on our modern dice with opposite sides summing to 7, but for some reason-possibly to acknowledge the holy trinity in hopes that they may have some influence- games were played with three dice. The games were simple: in mayores, he who rolled highest won; in other games, he who rolled lowest won.
Excerpted from What's Luck Got to Do with It? by JOSEPH MAZUR Copyright © 2010 by Joseph Mazur. Excerpted by permission.
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What People are saying about this
Persi Diaconis, Stanford University
Edward Packel, author of "The Mathematics of Games and Gambling"
David G. Schwartz, director of the Center for Gaming Research at the University of Nevada, Las Vegas, and author of "Roll the Bones: The History of Gambling"
Paul J. Nahin, author of "Digital Dice"
Meet the Author
Joseph Mazur is professor emeritus of mathematics at Marlboro College. His books include "The Motion Paradox: The 2,500-Year-Old Puzzle behind All the Mysteries of Time and Space" and "Euclid in the Rainforest: Discovering Universal Truth in Logic and Math".
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