Chances Are Page 8

  Diaconis and his colleagues looked at the shuffle as an extension of the random walk process to determine how many times you need to cut and riffle before all recognizable order is lost. The criterion they chose was the number of ascending sequences: if you had originally marked your pack from 1 to 52, could you still find cards arranged after the shuffle so that 3 was before 6 and 6 before 9—little clues that a cryptanalyst might use to deduce the original order? Surprisingly, there are quite a few of these sequences for the first three or four shuffles; in random-walk terms, Lucy stays very close to the blanket. It is only after six shuffles that the number of ascending sequences suddenly plunges toward zero. Only after seven, using this sensitive standard, is the pack truly randomized. This means that when you are playing poker at home and gather up the discarded cards to shuffle once, cut, and deal, you are not playing a game entirely of chance. You build your full house with the labor of other hands.

  Is there no place in a casino where the house relaxes its advantage? Are the gambling throng simply charitable donors, clubbing their pennies together to keep the owner comfortably in cigars and cognac? Well, the blackjack tables are a place where skill can gain a brief advantage over the house—but it is skill of a high order.

  In blackjack, you play individually against the dealer, drawing cards from a “shoe” and attempting to assemble a hand totaling 21points (or as close as possible to 21) without going over and being “bust.” The house advantage is simply that the player takes that risk of an overdraw first; as in Russian roulette, there is value in being second.

  The basic strategy in blackjack describes when you should “stand,” hoping the dealer either fails to match your total or goes bust; and when you should ask to be “hit” with another card. Here, the player has choice where the dealer must obey mechanical rules. So experienced players ask to be hit on a hand of 12 to 16 when the dealer’s first card is high, and they stand when it’s low. They double their bet (where allowed) on a hand of 10 when the dealer shows less than 9—and so on. There is a big, publicly available matrix of standard decisions in blackjack that help reduce the house advantage to an acceptable minimum.

  How, though, can the player reverse that advantage? By taking account of the one variable in the game that changes over time: the number of cards left in the shoe. Decks, when shuffled, are supposed to be random—but, as we saw in roulette, randomness can be lumpy. An astute observer, watching and keeping track of the cards as they come out of the shoe, can judge whether a disproportionately high or low number of top-value cards (10s and face cards) remain in the shoe. At this point, the player can modify the basic strategy to accommodate the unbalanced deck; the dealer, confined by house rules, cannot. After many deals, as the end of the shoe approaches, the careful card-counter has a brief moment of potential superiority, which, if supported by robust wagering, may lead on to moderate riches.

  An engineering professor, Ed Thorp, developed a computer program to recommend strategies based on the running count of cards; having applied its results successfully, he was banned from all casinos. The owners, recognizing the threat to the one immutable law of gambling—probability favors the house—reacted predictably. One-pack shoes were quickly replaced by two, four, or eight-pack shoes, so players would have to sit for hours, keeping track of more than three hundred cards, before their advantage kicked in. Nonetheless, if he were really willing to memorize several charts of modified strategy, mentally record a whole evening’s play, and give no indication to sharp-eyed pit bosses that this was what he was doing, a card-counter can still seize the edge for a deal or two—but anyone so skillful could make a better living elsewhere.

  As an optimist, you would say that life has no house edge. In our real battles with fate, an even-money chance really is even. That may be so, but probability theory reveals that there is more to a bet than just the odds.

  Let’s imagine some Buddhists opened a casino. Unwilling to take unfair advantage of anyone, the management offers a game at completely fair odds: flip a coin against the bank and win a dollar on heads, lose a dollar on tails. What will happen over time? Will the game go on forever or will one player eventually clean out the other?

  One way to visualize this is to imagine the desperate moment when the gambler is down to his last dollar. Would you agree that his chance of avoiding ruin is exceedingly small? Now increase the amount you imagine in his pocket and correspondingly reduce the bank’s capital; at what point do you think the gambler’s chance of being ruined equals the bank’s? Yes: when their capital is equal. Strict calculation confirms two grim facts: the game will necessarily end with the ruin of one party—and that party will be the one who started with the smaller capital. So even when life is fair, it isn’t. Your chances in this world are proportional to the depth of your pockets—the house wins by virtue of being the house.

  This explains why the people who appear most at home in the better casinos look so sleek, so well groomed, so . . . rich. They have lasted longest there, because they arrived with the biggest float. They alone have the secret of making a small fortune at the tables: start with a large one.

  There is another reason, though, for the natty threads and the diamond rings: gambler’s swank. Instinctively, the high roller understands the mathematics of gambler’s ruin: the smaller your bankroll, the sooner you’ll be heading out of town. The best strategy, then, is to play against other gamblers, rather than the house, and make your bankroll, real or apparent, work for you.

  Archie Karas is a man with a certain reputation in Las Vegas. Over six months starting in December 1992, he struck a remarkable upward curve, known in gambling circles simply as “The Run.” Starting at the Mirage with a borrowed $10,000, he tripled it playing Razz—seven-card stud in reverse, where the worst hand is best. Moving on to the pool tables, he cleaned out a high-rolling visiting businessman at eightball and then at poker; by now his bank was estimated to be around $3 million. Karas then challenged some of the best-known poker players in the world to take him on, individually and alone, at enormous stakes. These, too, he beat. In between matches, he would play craps at a table reserved for him—where his luck was slightly less good. By the end of the run, Karas, who had arrived with only $50 of his own money, was reputed to have won around $15 million.

  The Run is remarkable—and it illustrates many features of gambler’s swank. Karas was rarely matching wallets with the house; the most important phase was the extraction of the businessman’s money at pool, where Karas had perhaps a more realistic view of the probabilities than his opponent. Once Karas’ bankroll was in the millions, he could use it to intimidate at poker, blunting the skills of world-champion players with the sheer riskiness of playing at these stakes. Perhaps the best parallel is the story of one of the discoverers of statistics, Sir William Petty (a man we will meet again later). Desperately near-sighted, he was challenged in 1662 to a duel by the fire-breathing Puritan Sir Hierome Sanchy. What to do? How to balance such frightening odds? As the challengee, Petty had the choice of place and weapons—so he chose a blacked-out cellar and enormous carpenter’s axes. Sir Hierome demurred—proof that wit and gumption can be part of your bankroll as well as money.

  John Law was born in Edinburgh in 1671, son of a goldsmith in a country perpetually short of gold. Goldsmiths were Scotland’s bankers, in a pawnbroking sort of way, and the young Law was in a good position to see that trade and enterprise were being stunted by lack of capital. Economic theory of the time based the wealth of nations either on bullion or on a favorable balance of trade; Scotland had neither.

  When his father died, Law did what any young Scotsman with a legacy would do: he left. Traveling the Continent, he established a pattern of life that showed the most regular irregularity. He would arrive in a new city and take the best rooms. Splendidly dressed, tall, pale, and long-nosed, he would choose as mistress the local lady who best combined the virtues of beauty, birth, and availability. He would then set up the Game—at which he normally
acted as banker, and win all the money of anyone who cared to play.

  The card games of Law’s time were based on chance more than skill; they gave the banker only a small advantage. How, then, did Law win? Much as Archie Karas did. His front—the clothes, the rooms, the titled mistress, the icy calm—made him seem rich and indifferent to failure. Some of his success stemmed from his care and skill, but more from the apparent limitlessness of his funds: an opponent would be more likely to swallow a huge loss rather than return and risk the prospect of gambler’s ruin.

  Scotland, meanwhile, had suffered gambler’s ruin on the largest scale. Convinced that, lacking bullion, its only hope was an improved trade balance, the country risked all its wealth on an attempt to establish a trading colony on the swampy, mosquito-cursed coast of what is now Panama. It was doomed from the start: isolated and attacked by the stronger players, Spain and England, the little venture succumbed in months to famine and fever, quickly sinking back into the tropical ooze and taking every last coin in the kingdom with it.

  Why, reasoned Law, could not countries behave like him? If they wanted to play the game of nations, why did they need to put gold or exotic goods on the table first? After all, you don’t need to cash out until the game’s over—and the economic game never is. Chips—credit—that’s the proper medium for countries to gamble in.

  When Law arrived in Paris, the government of France was in the hands of a gambling regent, the Duc d’Orléans—and, though rich, its finances were in dreadful shape. Those who had funds hoarded them, so there was no money available for investment. Law proposed a solution based on credit: the Banque Royale, backed by state bonds, which would issue paper money, exchangeable for cash and acceptable for payment of taxes. These paper bills were a great success, so Law issued more, staking France’s new industries to their place at the table.

  As Archie Karas demonstrated, when you’re on a run, you don’t stick with the same game forever. Law diversified, acquiring the monopoly rights to trade with France’s Mississippi colony. He knew from Scotland’s experience that this would take a large investment to provide its potentially gigantic return, but he also knew that France was in the mood for just such investment. The Banque Royale issued shares in the Compagnie des Indes; investors needed to hold shares in the bank to buy shares in the company. An entire system of belief, of gambler’s swank on the largest scale, was marshaling and energizing the powers of the greatest nation on earth.

  The problem was that people believed in Law too much and assumed that shares in the Compagnie des Indes—so rare and so desirable—must be priceless. Across the land, people sold their chateaux, their fields, their beasts, and hurried to Paris to buy the magic stock. By the end of 1719, the market value of the company was 12 billion livres, while its income was barely sufficient to pay 5 percent on the nominal capital of one billion. And despite the pictures of friendly Indians handing over furs and emeralds in the company prospectus, Mississippi was still undeveloped forest and swamp.

  Law had attempted to win on a bluff; the apparent assets of his system—the government’s credit, the wealth of America, his own strange Northern financial genius—were the open cards, creating a belief that he held the perfect hand. As long as he could keep raising—issuing stock and bills that traded for more than their face value—he could win. He could run down the government’s debt and get the wheels of trade moving as well. But in liquidating all their assets, the French people saw him, raise for raise. He had met a bigger bankroll, and he lost.

  The end was terrible; in February 1720, the well-informed Duc de Bourbon drove up to the bank and redeemed his notes for a carriage full of gold. Panic quickly spread. There were terrible riots outside the Bank’s headquarters: duchesses, screeching for their lost investments, beat a way through the crowds with their fans. The shares of the company collapsed, pulling down the paper money of the bank, then the credit of the government. Heaps of the worthless banknotes were burned by the public executioner. Hard currency went out of circulation; Law had, for the moment, made France as cash poor as Scotland.

  Law failed, but gambler’s swank still reigns in finance. Why should your bank have chandeliers? Why does your stockbroker work in a marble and glass tower? Because they are playing on your behalf. True, there are the value investors who look for thrifty self-restraint in their advisors, but for every Warren Buffett lunching in a diner on a glass of milk and a tunafish sandwich, there are four fund managers glugging down Petrus ’49 and bellowing into bespoke cellphones. The room still believes that the big bankroll wins.

  There was once a magician in New York whose act showed indubitably why you should never play three-card monte (or Find the Lady) with people you meet on the street. He’d slap down the queen, point to it, shift around the three cards, lift the one we knew was the queen—and it wasn’t. It never was. He did the trick quickly; he did it slowly; he did it at glacial speed; he did it at glacial speed with cards the size of refrigerators. The switch was smooth and invisible; each time, we rubbed our eyes with our stupid hands and resolved to look more keenly at life.

  There is a demonstration in probability that has the same effect: you won’t believe it, you’ll be sure there’s a cheat involved, and it may make you very angry: the Monty Hall problem, named after the host of the television game show Let’s Make a Deal.

  You’ve been called down from your seat in the studio audience and now stand facing three doors: one conceals a large if unpopular car, the other two a goat each. Knowing fate’s indifference, you choose Door 1. The host now opens Door 3 and reveals a goat. “Now, you chose Door 1; do you want to stick with your choice—or switch to Door 2?” Your train of thought would probably go like this: “There were three doors available; now there are two. I don’t know what’s behind either door, so it’s an even split whether the car is behind Door 1 or Door 2. There’s no more reason to stick than to switch.”

  Probability takes a different view. When you chose Door 1 there was a 1/3 chance the car was behind it and a 2/3 chance it wasn’t. In opening Door 3, our host has not changed the original probabilities: there is still a 1/3 chance of its being behind Door 1—which means there is a 2/3 chance of its being behind Door 2! You are twice as likely to win the car (assuming you prefer it to a goat) by switching your choice than by sticking; and this is true no matter which door you chose in the first place.

  If you still find your mind boggling at the idea that opening one door doubles the chances for the remaining door, think of the situation this way. If the car is behind Door 1 (as it is 1/3 of the time), the host has a free choice of which other door to open; he is revealing nothing by opening one or the other. But 2/3 of the time—twice as often—he has no choice, since he must open the door with the goat, and by opening it, he is saying “The car is behind the door I may not open.” So his opening the door tells you something twice as often as it tells you nothing. The conservative strategy, the one backed by the odds, is to switch, not to stick with your original choice. Sometimes the unknown is less risky than the known.

  “Once, in Atlantic City, someone offered me a bet based on the Monty Hall problem.” Zia Mahmood is one of the most skillful players in world bridge, combining an uncanny card sense with an intuitive, bravura style of play. “I knew the answer had to be to switch. We use the same probability argument in bridge: we call it the Principle of Restricted Choice, where someone playing any one card of two suggests that he had no choice—two-thirds of the time.”

  Such is the human capacity for obsession and competition that the mere quarter-million possible five-card poker hands rapidly lose their intrinsic interest. Fortunately for those whose desire for complexity goes deeper, there is bridge. It offers 635,013,559,600 different possible hands. All are equally probable, although some are considerably more interesting than others; being dealt all the spades is just as likely as getting ♠ Q 10 9 5 4 2 ♥ 4 ♦ J 10 9 2 ♣ 9 2—the difference is that you feel the former is a sign of divine favor while the lat
ter is “my usual lousy luck.”

  We have come far enough together for you to be entitled to ask: “My chance of drawing all spades is one in 635,013,559,600? Just how do you come up with such a precise figure?” Well, spades are 13 out of the 52 cards, so your chance of taking a spade on the first draw is 13/52. If you did draw a spade, there will be 12 spades left out of 51 cards. So the chance of drawing two spades in succession is 13/52 × 12/51. Continue like this and you find your chance of drawing all and only spades is

  which when simplified is one in 635,013,559,600.

  Many odds calculations in bridge are similarly mechanical; the sort of deductive reasoning at which computers are particularly good—and there are, indeed, excellent bridge-playing computers now operating. Zia is convinced, though, that there is an intrinsic difference between the human and machine apprehension of probability in bridge because the uncertainties differ depending on exactly who is around the table. “Some players are machines, some Rottweilers, some are sensitive artists and some seem psychic. That means there is no one best play for a given situation; I could have the same hand and bid three hearts today and bid three spades tomorrow against different opponents—and have done the right thing both times.”

  The running horse seems to waken an old memory in all observers: roused by hoofbeats, the most prosaic soul resonates to the ancient themes of saga and romance. Horses are genuine—athletes who know neither transfer values, endorsement contracts, strikes, or silly hairstyles. The equine victor stands shivering in the winner’s circle, coat dark with sweat and veins bulging, gazing above our heads at something on the horizon; and we see the isolation of the hero, the being who has done what is beyond us—for us.