The New Math of Gambling

Today 47 states offer legal gambling. All you need to beat the house is some Internet savvy, the right number-crunching software, a thorough grounding in probability theory, and a few hours with an acting coach

By Brad Lemley|Monday, May 01, 2000


A side from his dancing left foot, anthony curtis, 42, looks like any other sucker tourist dropping a wad at the Binion's Horseshoe blackjack tables in Las Vegas. Hair slicked back, nose busted twice from rugby, lean as a desert lizard, left hand curled around his second imported beer--not water, a dead giveaway for card-counters--and with a thin but frequent smile, he seems wound tighter than most, maybe, but inside the bounds--could be a killer lawyer or military. After three minutes of play, he's down $100. He's flirting with the dealer ("Why are you so friendly? The other dealers here are all crabby"), seems distracted by the waitress's three inches of cleavage, and acts surprised when his luck turns. "My God, this never happens," he murmurs as he flips a natural 21: a 10 of hearts and ace of diamonds. Next two hands, he stands on 17 and 18; the dealer, whose demeanor is actually quite dour, busts both times. Now he's up $150. "Better quit," he says, clumsily scooping up 12 green chips and striding to the cash-out cage. The pit boss--whose job it is to spot counters--smiles and waves at him. The dealer doesn't even look up.

While Curtis did his dumb-luck shtick above the table, the real action was below, down on the gaudy carpet. One of the best blackjack players in the world--he won the World Match-Play Blackjack Championship in 1987--Curtis counted every card played, maintaining a multiparameter tally, which means he tracked several things at once. His left foot's job was to tick off the aces. When one ace had been dealt, he put his heel down; second ace, he tilted his foot to the left; third, up on his toe; fourth, tilted to the right. Meanwhile, in his head, he kept a running count of the other cards: Twos and eights each counted as plus one; threes, fours, sixes, and sevens were each plus two; fives were plus three; nines were minus one; and tens and face cards (which all count as 10 in blackjack) were minus three. The idea is that blackjack--unlike, say, craps--is a game of dependent events, and tracking the dealt cards yields valuable information about the ones that remain. A wealth of tens and aces left in the deck favors the player, because the player is more likely to get a blackjack (for which the house pays 11Ú2 times the bet) and because the house, which plays by fixed rules, is more likely to bust. Whenever Curtis's count went positive--that is, above zero--he boosted his average bets and, on this evening, his winnings. Do it all while looking like a flirtatious, distracted, none-too-bright vacationing drunk, and you, too, can win big.

Card-counting can be daunting--it's a science that's under constant revision and an art that demands great discipline--but it can also be extremely rewarding. By 1989 Curtis had won enough money at the tables to bankroll a gambling-book publishing house. He still plays in his off-hours and remains plugged into the professional blackjack subculture.

Standing shoulder to shoulder with the hapless hordes, Curtis represents the tiny coterie of mathematically talented, driven, inconspicuous gamblers who methodically drub the casinos at their own games. These players operate at each wagering stratum, from low-rollers who use computer-crafted strategies to scoop $15 an hour from video poker to mathematicians who accumulate monumental winnings. Armed with computers and strategizing via the Internet, modern players are bringing unprecedented sophistication to the ancient business of vivisecting gambling games in order to extract an edge. Even those who don't plan to go pro can learn from the new gambling elite and turn a potentially impoverishing vacation into a profitable experience. "To figure this stuff out is just very alluring to someone like me," says Olaf Vancura, a casino-game creator with a Ph.D. in physics and the author of Knock-Out Blackjack, a new card-counting system that's within the reach of the average intellect. (See "Card-Counting Made Simple," page 64.)

Winning over the long haul demands research, stamina, nerve, and no small measure of acting ability, but the principal requisite is a rock-solid grounding in probability. Curtis estimates that 99 percent of what's offered in casinos and other legal betting venues is what he terms negative expectation games, meaning that over the long haul, the house will harvest anywhere from 1 percent to 60 percent of money gambled. Americans who failed to appreciate this fact lost $55 billion in legal wagering in 1998, or about $203 for each man, woman, and child. Losses have mounted every year for the last two decades, largely because legal gambling has exploded in that period. Today, all but three states--Hawaii, Utah, and Tennessee--offer legal gaming in some form. In 1998, 63 percent of Americans reported having gambled in the previous year.

The average bettor accepts his skinning because he imagines the house odds are insurmountable. The average bettor is wrong. People always say, "Who pays for the lights in Las Vegas?" observes Jean Scott, a 61-year-old video-poker-playing grandmother who, with her husband, Brad, earned enough in two years to purchase their Las Vegas condo. "All I know is, I don't."

The next afternoon, Curtis chain-drinks coffee in his windowless office in a one-story stucco building, a half-block north of the 51-story Rio casino with its red-purple neon stripes. He wears a knife-creased white shirt, pointed black-leather shoes, and a thin wristwatch, hunching forward when he talks to reveal gnarled cords in his neck, a legacy from his days as a varsity college wrestler, 142-pound class. He makes relentless eye contact.

This is the headquarters of Curtis's publishing house, the Huntington Press, which aims to have the same relationship with the casinos that a tick has with a dog. Stacked all around are teetering piles of the gambling-math books he sells, with titles like Burning the Tables in Las Vegas, The Theory of Blackjack, and Casino Secrets. He waves a hand at them.

"I was born to do this," he says.

Growing up in Detroit, a fanatical game player since age 4, he found his calling at 16 when a family friend gave him Charles Einstein's How to Win at Blackjack. "I read it. I understood it. I read more books. I made my decision at that age. I would go to Vegas and play games for a living."

The books that inspired Curtis grew out of a flowering of scientific gambling analysis that began with "The Optimum Strategy in Blackjack," an article printed in the September 1956 issue of the Journal of the American Statistical Association. Four Army mathematicians, known today as the Baldwin group, demonstrated how to play each hand to maximize winning potential, proving that a player who followed their program in a single-deck game would, over time, be virtually even with the house. That made blackjack by far the most advantageous casino game to play. (American roulette, by contrast, typifies the negative-expectation gambling game. Consider a bet on red. Because the wheel has green zero and double-zero slots, fewer than half--in fact, just 18--of the wheel's 38 slots are red, but red wagers pay 1 to 1. The probability of a win is 18/38; of a loss, 20/38. The expected payoff is (1 x 18/38) Ð (1 x 20/38), which equals Ð2/38, or Ð.053, or Ð5.3 percent. In other words, over the long haul a gambler will lose $5.30 for each $100 wagered.)

But the giant of blackjack theory, and nearly a god to young Curtis, was MIT mathematics professor Edward O. Thorp, who published Beat the Dealer in 1962. The Baldwin group had focused on how to optimize play using each hand's visible cards, reckoning that card-counting was impractical. Thorp's twin breakthroughs, bolstered by then-nascent computer calculation, were that smart folks could learn to count cards as they were played and that, in doing so, these gamblers could gain a consistent advantage over the house.

On his own, day after day, Curtis practiced card-counting until his brain ached and his fingers were raw. At 21, he turned his back on a UCLA wrestling scholarship, moved to Las Vegas, bellied up to the Sahara's blackjack pit, and put his $1,800 life's savings on the line. "The first night, I won $22. The second day, I lost $900. At the table, I started hyperventilating. Then I got physically sick."

Curtis had slammed into the brutal reality of fluctuation around the average return, or what statisticians call standard deviation. Although an astute blackjack card-counter can, over the long haul, realize up to a 1.5 percent profit (depending on such variables as the number of decks, which card-counting system he employs, and how many cards remain before the dealer reshuffles), losing streaks can last for weeks. There's a saying: If you can't stand the flux, don't bet the bucks. Many technically proficient blackjack counters have been run out by flux.

An elegant mathematical expression of standard deviation's power in gaming is the so-called risk of ruin, which establishes the likelihood that a particular bankroll will go to zero before it doubles. According to calculations by Vancura, a blackjack card-counter playing with a 101.11 percent expected return (the expectation of one version of his card-counting system) and a bankroll that's just 25 times the minimum bet has close to a 47 percent chance of going broke before doubling his money. To drop the likelihood of going broke to, say, .5 percent requires a bankroll 1,000 times larger than the game's minimum wager; in a $5 to $25 game, that's $5,000. Chastened, Curtis returned to the real world, first as a stockbroker, then as the smallest bouncer in Las Vegas.

The lean years ended when Stanford Wong--the nom de guerre of a statistics and finance professor who relishes anonymity--enlisted Curtis to play tournament blackjack. Financed by Wong, Curtis was free to make the big bets that would harvest big money. After he retired from tournament play, Curtis set up his publishing house. Today, while still winning a sizable return in his off-hours, he reigns over a stable of authors in his growing library of gambling science. Among them is a rising star named Jean Scott.

Jean Scott's Fibromyalgia inflicts chronic pain in her arms and shoulders, so to play video poker, she sits and props her feet on either side of the machine's base. This lets her rest her forearms on her bent knees and spread her fingers on the BET, DEAL, DRAW, and hold buttons just as a typist would on a keyboard. Thus, she plays 600 hands and cycles $3,000 through the machine each hour. Bathed in the screen's hypnotic glow, she is rapt.

Scott, a minister's daughter, was raised to believe that gambling is the devil's snare. She didn't learn the names of the four suits until age 35, but she has vigorously made up for lost time. From her gold-lamŽ slippers, to her dice-motif stickpin, she exudes Grandma-gone-Vegas. "Even as a kid, I played Chutes and Ladders for blood," she says. Now, as author of The Frugal Gambler, which deconstructs video poker for profit, she proffers a justification for her late-in-life rebellion. "I just say to people, I'm doing God's work. I'm taking money away from those awful casinos."

Video poker, the blinking, beeping simulacrum of the venerable card game, was Scott's vehicle to success. Today, she is playing in the Orleans, an off-the-strip casino favored by Las Vegas residents (first rule of successful gambling--patronize casinos that cater to locals, who demand more statistically advantageous games). The machine she is playing is, like most, based on five-card draw: The player discards the worst of five cards dealt by the machine and draws replacements in hope of creating a winning hand. Tables printed on the front of each machine list payouts: In a $5-per-play machine, a jacks-or-better pair might simply return the wagered $5, while a royal flush might pay $4,000.

Scott scouts for promising machines and writes their payouts and their rules on a notepad. (For instance, a machine that pays nine times the bet for a full house, rather than the typical eight times, may be worth more of her time.) Then she goes home and punches the information into a computer program called Win Poker. Such programs play millions of hands to determine the odds of achieving a given winning hand and compare that to the machine's payout schedule. In seconds, the calculated ratio of odds-to-payout yields two crucial facts: which outcomes to play for in order to maximize a particular machine's return and precisely what that return will be. (While managers don't permit computers in casinos, there is no injunction against bringing in a printout that lists optimum play.)

One of Scott's favorite machines, the one she is milking today, is called Deuces Wild. Played perfectly and long enough, it returns 100.76 percent of the money put into it, or $22.80 profit per hour at Scott's 600-hands-per-hour pace. (She concedes that perfect play is unlikely and has determined that her typical error rate drops the profit to .5 percent, or $15 per hour.) See here, she says, pointing at the display, which shows a ten of spades, ten of hearts, king of hearts, king of spades, and jack of clubs. The intuitive play would be to discard the jack of clubs and go for the full house. Instead, she discards the tens and the jack, going for four of a kind. Typical video poker players employ the same strategies they would use while playing cards with friends around the kitchen table. That's why they lose, Scott says. Video poker consists of a single hand played against a machine, so there are no other hands to consider, no bluffing, no personalities. Statistically, on this particular machine, a full house occurs every 47.1 hands and pays three times the bet, while four of a kind occurs once every 15.39 hands and pays five times the bet. Four of a kind is, then, the meat of this machine--a high payer relative to its frequency. In fact, of the machine's 100.76 percent optimal-play return, 32.47 percent comes exclusively from four-of-a-kind hands.

The most common playing mistake, Scott says, is trying too hard for royal flushes, the big payout. The mark of an odds-savvy video-poker player is that he follows the computer's mandates religiously, capturing a steady stream of moderate payouts rather than the dramatic huge ones. Scott emphasizes that fluctuation matters in video poker just as it does in blackjack. (In a bad month, she and her husband Brad have lost $10,000.) For a greater-than-100-percent return to manifest in any video poker game, the player must play long enough to get a royal flush--because that royal flush contributes about 1.7 to 2 percent of the total payback percentage--without going for the royal flush in low-percentage situations. Statistically, on the Deuces Wild a royal flush occurs only once in every 40,390 hands. That's the equivalent of 67 hours of play at 600 hands per hour, or eight hours a day for more than eight days.

Scott and Brad--a lifelong gambler who's as quiet as Scott is talkative--don't play that much, averaging five hours a day between them, reaching the requisite 67 hours in about two weeks. They are retired. "I happen to love video poker, so I like to play every day," says Scott. "But for some people, it can be too much like work."

All of this can, and should, give one pause. few of these winning techniques are top secret. As Scott points out, anyone who can read her book or operate a computer could learn to make a decent income in the casinos. And the casinos certainly have the knowledge and power to snuff out the odd beatable game or, more to the point, never offer it in the first place. But winnable games keep appearing, and casino management maintains a curiously sanguine attitude toward the proliferation of crush-the-house books, software, and seminars.

The reason for the complacency can be grasped by revisiting the turbulent Las Vegas of the mid-1960s. On March 27, 1964, Life magazine predicted Thorp's Beat the Dealer would foment a players' revolution, proclaiming, "Thorp does not cheat. But Thorp cannot lose." Casino managers panicked, anticipating swarms of card-counters who would rake in chips like clams at low tide. On April 1, 1964, most Las Vegas casinos changed the rules of blackjack, the first and only time that the rules of a major casino game had ever been radically altered. A pair of aces could no longer be split, and doubling down was restricted to two-card totals of 11. In gaming terms, the changes were roughly equivalent to depriving a chess player of a rook. Players abandoned the tables in droves.

Desperate, casino managers nervously reinstated the old rules. Players rushed back to the tables, and managers relaxed once they realized that the house percentage-per-player take was just as large as ever. In fact, the casinos made more money, because more people were drawn to the beatable game. It's not enough just to show up at a game that can be beaten, says Vancura. One also needs to play properly. Most people are unwilling or unable to learn the system.

Even if Vancura's simplified card-counting book achieves best-seller status, as Thorp's did, the lights will likely keep blazing just as fiercely in the desert and along the boardwalk and in the Indian casinos and on the riverboats and aboard the ocean liners. Curtis, Scott, and Vancura all seem appalled at the apathy and ignorance of the average gambler. Successful gambling is unquestionably possible, and someday, they hope, the fleeced hordes will decide they've lost enough. Someday casino profits may shrink from the obscene to the merely respectable, and the successes of those who win through intellectual effort rather than luck may inspire a newfound appreciation for the beauty, wonder, and irresistible power of probability.

But don't bet on it.
Card-Counting Made Simple

Since the 1950s, mathematicians have hatched dozens of blackjack card-counting systems, seeking a delicate balance: easy enough for the average person to use, powerful enough to offer a significant edge. The search may be over. In their book Knock-Out Blackjack, casino-game designer Olaf Vancura and electrical engineer Ken Fuchs offer what may be one of the easiest and strongest professional-level systems ever published. In a single-deck game, the system, executed perfectly, will return 101.53 percent of money wagered.

All counting systems require the player to assign number-values to the cards played, but the K.O. system keeps the numbers simple: Cards two through seven are assigned a value called plus 1; eight and nine are assigned the value zero; and the ten, jack, queen, king, and ace are all assigned minus 1. With practice, it becomes easy to glance at a plus-1 and minus-1 hand--say, a seven and a jack--and cancel it, says Vancura. As a practical matter, then, only the tally from a few cards must be remembered between hands.

Many systems also require side counting, independently tracking the appearance of one card, typically the ace. The K.O. system requires no side-counts.

Finally, in multideck games, many systems require the player to divide the so-called running count--the tally of the numerical values of cards played so far--by the estimated number of decks remaining. This determines the true count, which dictates betting decisions. Example: The running count is plus 3, and an estimated 11/4 decks have been played in a two-deck game; divide plus 3 by the remaining 3/4 (deck), round toward zero (when necessary), and derive the true count: plus 4. That means it is a good time for a sizable bet. The K.O. system requires only a running count; no need for a true count.

Like most modern gambling innovations, this method is an artifact of computer power, based on simulation software written by Fuchs and Vancura that can play 200 million games per hour. While Vancura uses it for recreation, "I don't feel compelled to do it full time. For me, the real fun is figuring it out." --B.L.



Many thanks to the Fiesta Casino Hotel in Las Vegas for allowing us access to the casino while researching this story. Known as the "royal flush capital of the world," the hotel is a mecca for video poker players because it pays out on an average of 100 royal flushes a day. The casino's Web site is www.fiestacasinohotel.com
Stanford Wong's Current Blackjack News, Stanford Wong, Pi Yee Press, monthly. Hungry for business, casinos often make short-term blackjack rule changes that favor the player, such as paying 3 to 1 for naturals (two-card totals of 21) as opposed to the typical 1.5 to 1. Reporters around the nation ferret out such advantageous situations for this newsletter, available on-line for $12.25 per quarter; details at www.bj21.com

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