The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse
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He starts with basic protocol: how one handles the dice. When a new craps table opens, for instance, the dealer opens a fresh, factory-sealed pack of five dice, from which the inaugural “shooter” must select two. “Whatever you do, don’t grab all five dice, toss them across the table, and yell, ‘Yahtzee!’ ” Dominic cautions. Then everyone will know you’re a rube.
The dice must be held in one hand, to prevent players from surreptitiously switching in loaded dice. You aren’t allowed to rub the dice between both hands for the same reason, or kiss the dice (“You don’t know where they’ve been,” quips Dominic), and while it’s fine to lightly blow on the dice for good luck, we were advised not to get spittle on them, out of courtesy for the next shooter. By order of the Nevada gaming commission, the casino also requires that both dice bounce off the far wall of the table on each roll, lest certain players try to “rig” the roll. We all take turns practicing this. Dominic warns us not to throw the dice too hard, but that doesn’t stop one overexcited shooter from tossing them so forcefully that they bounce off the table—narrowly missing a drop down a buxom brunette’s cleavage.
THE DUKES OF HAZARD
Some version of craps has been around for centuries, although historical accounts quibble over the details. Did craps derive from an old game called hazard, popular with English knights during the Crusades as they laid siege to a castle called Hazarth in 1125 A.D.? Perhaps the game is Arabic in origin (al-zar in Arabic translates as “the dice”). Or does craps reach further back in history to the Roman Empire, when soldiers fashioned rough-hewn dice out of pig knucklebones? There are certainly references to the game in Chaucer’s Canterbury Tales, and it was hugely popular in France by the seventeenth century, especially among the aristocracy.
We can credit a French-Creole American nobleman with the tongue-twisting moniker of Bernard Xavier Philippe de Marigny de Mandeville for bringing craps to America. The son of a count, Marigny was born to wealth and privilege on the family’s New Orleans plantation in 1785, and his upbringing did much to foster a sense of entitlement. Local lore tells of the 1798 visit to the Marigny estate by the Duc d’Orléans, Louis-Philippe (later crowned king of France in 1830), and his two brothers, and the lavish revels that ensued, including the manufacture of special gold dinnerware. In a show of excessively wasteful extravagance, the gold place settings were purportedly tossed into the river when the festivities ended, for no one would be worthy to eat from any plate used by Louis-Philippe. (One hopes the poverty-stricken locals trawled the river bottom and scavenged the discarded loot.)
With such role models before him, it is small wonder that young Master de Marigny matured into a spoiled, dissolute, and extravagant young man, coming into his substantial inheritance at the tender age of fifteen after the death of his doting father. His long-suffering guardian despaired of controlling the headstrong teenager and shipped Marigny off to London in hopes that there he might learn some temperance. Instead, Marigny frequented any number of gambling dens, most notably the infamous Almack’s. That’s where he learned the game of hazard, bringing a simplified version of it back home to New Orleans a few years later. In local dialect, the game was dubbed crapaud, from a derogatory term for the French in New Orleans, Johnny Crapaud. English-speakers later shortened the name to craps.20 The game quickly spread to the Mississippi riverboats and beyond.
Bernard de Marigny died penniless in 1868, having repeatedly subdivided his once vast plantation into numerous land parcels, selling them off to cover his ever burgeoning gambling debts. He is largely forgotten, but two legacies remain: the Faubourg Marigny neighborhood of New Orleans (built on the site of the old Marigny estate), and the game of craps, which is more popular today than ever. In fact, there is a street in the Faubourg Marigny district named Craps, reflecting its founder’s place in gambling history.
There have been many refinements to the rules of craps over the centuries, but the fundamentals remain unchanged. Each player takes turns being the shooter, rotating around the table as each individual game comes to an end. Every game starts with a “come-out” roll: Players place their initial bets on the “pass” line (required in order to play), and the shooter rolls the dice. If the shooter rolls a 7 or 11, everyone who placed a pass bet wins outright. If the shooter rolls a 2, 3, or 12, everyone loses outright. If the shooter rolls any other number, that number becomes the “point” for the duration of the game.
Our first shooter is a middle-aged man of Eastern European descent—let’s call him Yuri—visiting Vegas with his wife. He starts off strong, rolling a 7 right off the bat, and the table cheers in victory. We collect our winnings, place new line bets, and Yuri rolls again. It’s an 8; this becomes the point, and the game’s afoot. Now that a point has been established, we keep betting and Yuri keeps rolling until he rolls the point again (an 8) or he rolls a 7 (craps). If the former, we win; if the latter, we lose. Either way, the game ends, a new shooter takes over, and the cycle begins anew.21
If that were all there were to craps, it would become boring very quickly. So as the game evolved, additional types of bets were added, each with its own set of odds. For instance, as an alternative to the standard pass bet on the come-out roll, a player can place a “don’t-pass” bet—essentially betting against the shooter and the rest of the table. One caveat: This will make you very unpopular. It’s a very social game, and players tend to bond at a craps table, because people’s fortunes rise and fall with the shooter’s. Betting against the shooter is a buzzkill. For don’t-pass bets, the win-lose rules are reversed. If the shooter rolls a 2 or 3, a don’t-pass bet will win outright, while the rest of the table loses. If the shooter rolls a 7 or 11, a don’tpass bet will lose outright—and everyone at the table who placed pass bets will revel in Schadenfreude.
The key difference is if the shooter rolls a 12. In that case, a don’t-pass bet will neither win nor lose; it would be a “push.” This is simply a means of maintaining the house advantage: Three numbers are losers while two are winners on the come-out roll if you place a pass bet. In contrast, two numbers are losers and two are winners if you place a don’t-pass bet on the come-out roll. One might be tempted to conclude, therefore, that the odds of winning the come-out roll with a don’t-pass bet are 50/50. One would be mistaken. Probability is more complicated than that, even for a relatively simple game like craps, which is why the field has fascinated scientists and mathematicians for centuries.
CHANCE ENCOUNTERS
Among the first to analyze games of chance with an eye toward odds and winning strategies was a sixteenth-century physician, astrologer, and mathematician named Gerolamo Cardano. Born in 1501, his was not the most auspicious of beginnings. His mother, having already borne three children and clearly being fed up with parenthood, tried to abort him with a brew of wormwood, barleycorn, and tamarisk. Gerolamo survived but promptly contracted bubonic plague when he was just a few months old—usually a death sentence at the time, particularly for an infant. Astoundingly, he survived that, too. (His wet nurse and three half-brothers perished.)
His father, Fazio, wanted the teenage Gerolamo to study law, but the boy longed to study medicine instead. He initially supported his studies by tutoring others in geometry, alchemy, and astronomy, as well as casting horoscopes. (Astrology and alchemy were still considered legitimate fields of study.) But then he developed a taste for gambling and found he had a gift for beating the odds. He quickly amassed winnings of 1,000 crowns, more than enough to pay for his education, and in 1520 began writing a treatise, The Book on Games of Chance, which he kept revising right up until his death.
Cardano was a better gambler than a physician, it seems—or rather, he lacked the business acumen to market himself to prospective patients. He struggled mightily to support his family early in his career, and soon found himself resorting to gambling again to make ends meet. Eventually Fortune seemed to smile on him: He published a series of successful books and by 1550 became the renowned physician he’d always dre
amed of being.
If only he hadn’t had children. Cardano’s appalling offspring were a trio of bad seeds whose behavior would make Caligula blush. His daughter Chiara seduced her older half-brother, Giovanni, at the age of sixteen, became pregnant, aborted the fetus, and continued to philander even after her marriage, eventually contracting syphilis. That same brother was later convicted of poisoning his wife; Cardano spent a fortune on his defense, to no avail. Giovanni was summarily executed, most likely deservedly so. The younger son, Aldo, became a torturer for the Spanish Inquisition, testifying against his own father so that Cardano briefly landed in jail. Cardano finally died in September 1576, penniless and quite mad, having burned more than half of his manuscripts before shuffling off this mortal coil.
Among his surviving manuscripts was The Book on Games of Chance, finally published in 1663, almost a century after Cardano’s death. By that time, others had replicated and out-paced Cardano’s analysis, but the beleaguered physician with the rotten luck deserves his minor place in the annals of probability theory. In chapter 14, titled “On Combined Points,” Cardano laid out what we now know as the law of the sample space. The sample space is simply the set of all possible outcomes of a random process (like the roll of the dice or flipping a coin). Cardano reasoned that the probability of winning a roll of the dice, for example, is equal to the proportion of winning outcomes. A die can land on any one of its six sides, and those six potential outcomes make up the sample space. Place a bet on one such number, and your chance of winning is 1 in 6; place bets on three such numbers, and your odds improve to 3 in 6.
His methodology served him well as a gambler, but Cardano’s analysis was rather flawed. He assumed that all outcomes were equally likely; in fact, different outcomes have different probabilities. Galileo Galilei demonstrated this in the early seventeenth century in a short paper entitled “Thoughts About Dice Games.” Galileo wasn’t especially interested in probability theory, preferring to roll balls down inclined planes and time their rate of travel. But his patron, the Duke of Tuscany, was an inveterate gambler and thus keenly interested in the question of why—for games played with three dice—the number 10 seemed to occur a tad more frequently than the number 9. Galileo concluded (correctly) that this occurred because there were more combinations that yielded a 10 than yielded a 9. There are twenty-seven ways to roll a 10 with three dice, compared to twenty-five possible combinations for a 9. It’s now an established tenet of probability theory that the odds of a particular outcome are dependent on the number of ways in which it can occur.
Galileo took his analysis no further; his research interests lay elsewhere. Yet wealthy and titled patrons with gambling problems continued to push for advances in probability theory, most notably a social-climbing French essayist named Antoine Gombaud, who adopted the title chevalier after the character in his many dialogues who represented the author: Chevalier de Mere.
Gombaud was a man of letters who fancied himself an amateur mathematician, and in 1654 he found himself pondering what is known as the problem of points: How do you determine how the stakes in a game of chance should be divided if, for some reason, the players were interrupted and never finished their game? It was first proposed in 1494 by an Italian monk named Luca Pacioli in his treatise Summa de arithmetica, geometria, proportioni et proportionalita. (Yes, even monks fell victim to the lure of gambling. They didn’t have television in the Middle Ages.) So this question had been knocking around gambling circles for nearly two hundred years by the time Gombaud decided enough was enough—he wanted a solution to the conundrum.
Gombaud turned to a young mathematician named Blaise Pascal, who had taken up gambling when his doctors advised him to abandon mental exertions for the sake of his health. Pascal suffered from chronic stomach pain, nausea, migraines, and partial paralysis of the legs, among other ailments. Intrigued, Pascal quickly realized he would need to invent an entirely new method of analysis to solve the puzzle, because the solution would need to reflect each player’s chances of victory given the score at the time the game was interrupted. Thus began his legendary correspondence with fellow mathematician Pierre de Fermat, which over the course of several weeks, laid the foundation for modern probability theory. They quickly realized that in order to solve the problem it would be necessary to list all the possibilities and then determine the proportion of times that each player would win.
Caltech mathematician Leonard Mlodinow gives one of the clearest explanations of how to solve the problem of points in his book The Drunkard’s Walk, using the example of the 1996 World Series, in which the Atlanta Braves beat the New York Yankees. Atlanta won the first two games, but what were the odds of a Yankee comeback at that point? To get the answer, you would need to count every scenario in which the Yankees could have won and compare that to the number of scenarios in which they could have lost. By that reckoning—which assumes that the Yankees and the Braves had equal chances of winning each subsequent game—the chance of an overall Yankee victory would have been 6 in 32, or around 19 percent, compared to 26 in 32, or about 81 percent, for an Atlanta victory. “According to Pascal and Fermat, if the series had been abruptly terminated, that’s how they should have split the bonus pot, and those are the odds that should have been set if a bet was to be made after the first two games,” Mlodinow concludes.
Pascal’s bank account may have suffered during this period of his life, but his health was never better. Ironically, the mental exertions of his correspondence with Fermat triggered a “trance” a few weeks later, from which Pascal never fully recovered. He became deeply religious, eschewing his former “corrupt” ways, and died of a brain hemorrhage at thirty-six. Maybe he should have stuck with gambling.
RISK AND REWARD
What happens in Vegas stays in Vegas, or so the advertising tagline goes—and more often than not, your money stays too. For craps, the probabilities are fairly straightforward because there are only two dice with six sides each, so there are only 36 possible combinations: six possibilities for each of the two dice (6 × 6 = 36). Yet not all outcomes are created equal, and therein lies the secret of the house advantage. There are more ways to roll a 7 than a 2, for example. To roll a 2, you would need to roll snake eyes (1 + 1). In contrast, there are three different combinations that total 7: 1 + 6, 2 + 5, and 3 + 4. Furthermore, because each die is distinct, probability also includes the combinations 4 + 3, 5 + 2, and 6 + 1. Ergo, 7 is the most likely number to be rolled. It’s no accident that the losing roll (craps) is 7 once the game gets under way.
How does this play out when one takes into account the odds and payoffs for the various bets in craps? For the pass and don’t-pass bets, the payoff odds are one to one: Winners receive one dollar for each dollar they bet. This does not mean you have a 50/50 chance (0.5) of winning a pass-line bet; the actual probability is exactly 0.49293—just slightly less than a 50/50 chance, giving the house an edge of about 1.42 percent.
Once the point has been established, the next most favorable side bet to further narrow the house’s edge is called a “freeodds” bet. For example, before the next roll, S-Money would add between one and three extra chips behind his original pass bet. While line bets have a one-to-one payoff, the payoff for a free-odds bet is determined by the exact mathematical odds against winning the bet. If the point is either a 4 or a 10, the odds against winning are 2 to 1; ergo, the payoff if the shooter rolls the point before a 7 is 2 to 1. So if S-Money bet $10 as a free-odds bet, he would win $20. If the established point is a 5 or a 9, the odds against winning are 3 to 2, so the same $10 free-odds bet would win $15. And if the point is a 6 or an 8, the odds against winning are 6 to 5, so a $10 free-odds bet would bring in $12.
Dominic assures us that the free-odds bet gives the house the smallest possible advantage, and thus is an excellent way to maximize one’s winnings. But there’s still just a single number (the point) by which you can win. You could increase your chances of winning if there were more possible winning numbers. Th
at’s where the “come” bets and “point” bets come in. For a come bet, you place a chip in the come section of the table before the next roll, and whatever number the shooter rolls becomes a new point for that particular chip. So now you can win on this point or on the original point established on the come-out roll. If the shooter rolls a 7, of course, you lose outright—because the game is over.
There are some disadvantages to the come bet. First, it is a “contract bet,” meaning it remains in place until the end of the game, just like the pass and don’t-pass bets. (You can also place don’t-come bets, once again betting against the shooter.) Second, you are at the mercy of a roll of the dice to determine the new point. If you want to be able to pick your own number to be the new point, and have the freedom to add or remove chips at will, you should make a point bet. The payoff odds aren’t quite as good as the come bets, but you have more control over the board. There are plenty of other, higher-risk bets, but these are really the only bets you can make in craps with reasonable odds. You’ll still lose money in the long run, but you’ll lose it much more slowly.
Now it’s time to play the game for real. The casino graciously sets up a small-stakes table just for the newbies for one hour so we can practice our newfound skills. S-Money being the more math-savvy in our household, he shrewdly opts to take Dominic’s advice and maximize the size of his odds bets in relation to his line bet, thereby reducing the house edge to a whisker (although not eliminating it entirely). Because we’re experimenting, I choose to focus on placing come bets and point bets, trading better payoff odds for more control over the board, to see how these two strategies compare.