Chances Are Page 7
If this were all there is to roulette, of course, the casinos would simply be charities redistributing wealth from the proponents of red or odd to black or even and back again. The little extra that earns them everything—is nothing: the zero that joins the 36 numbers on the wheel. It’s not a magic number—you may bet on it if you wish—but its presence subtly shifts the alignment of payoff and probability. Your bet on any one number pays 36 times, but the event happens only one time in 37. Your cautious chips on rouge or impair may double your stake; but either chance will come up, not half the time, but half minus 1/74.
Why would anyone possessed of his senses put even money on an uneven chance? Who would volunteer to pay a tax of 2.703 percent—for that’s what the discrepancy between chance and payout represents? Some see it as a fair fee for an evening’s entertainment; it’s like going to have one’s sense of risk massaged. Others look more closely at the whole proposition and—by deconstructing the idea of probability it represents—manage occasionally to make a killing.
Joseph Jaggers was a 43-year-old no-nonsense Lancashire cotton-spinning engineer; a man proof against gambling’s seductive powers. When he arrived in Monte Carlo in 1873, he knew nothing about roulette—but everything about spindles. As laymen, we assume that the wheel, so silent and smooth, must spin true. Jaggers’ professional life, though, was an unavailing struggle to make things spin true. There is no earthly mill where all hums perfectly on its axis. Machinery shares man’s imperfections: it wobbles and chatters—to a practiced ear, intelligibly.
Jaggers hired six clerks to record every number generated at all the casino’s tables for a week; he locked himself in with the results and found what he sought: one wheel had a distinct bias. The next day he arrived at the tables and began to play his advantage. After a day of heavy losses, the management switched the wheel to another table. But Jaggers was observant: puzzled at losing the next morning, he remembered a tiny scratch on “his” wheel that was no longer apparent. He looked for it among the other tables and was soon reunited with the untrue spindle. In all, he won the equivalent of $325,000 and then returned to Lancashire, never to gamble again—although in truth he had never been gambling at all.
The ball must land somewhere. The cup in which it finally nestles is the solution to a complex but mathematically determined sequence of collisions, each with its own precise possible description in terms of the angular momentum of its components. We call this solution “chance” because we assume that these collisions are complex enough and happen sufficiently quickly to make physics throw up its hands and accept that the only moment at which we can understand what has happened is after the sequence is finished. Yet if you could rewind time, the last bounce or two might be less difficult to predict than the whole sequence, assuming that you could gauge accurately the speed and spin of ball and wheel at that point. This was thought to have been the technique of Charles Deville Wells, The Man Who Broke the Bank at Monte Carlo.
Claude Shannon, MIT’s deviser of information theory, thought the same computers that solved the equations of missile trajectory could master the wheel, so he built one in his basement. Norman Packard and Doyne Farmer followed in his footsteps; their 1980 shoe-mounted computer, the Eudaemonic Pie, attempted to solve the equations of motion for those last moments between the slowing of the ball and the closing of the bets. Despite its puny 4K processor, it worked too well for the comfort of Las Vegas casino owners; it is now a felony in Nevada to enter a gaming place with a computer in your shoe. Roulette’s simplicity, its plain-sight mechanism, provides a constant temptation to seek out an eddy of determinism in the flow of randomness.
All this, though, is to miss the point of gambling, which is to accept the imbalance of chance in general yet deny it for the here and now. Individually we know, with absolute certainty, that “the way things happen” and what actually happens to us are as different as sociology and poetry. Our personal story has meaning: what happens to us is fate, not mere occurrence. We challenge destiny at the tables; and when we climb those carpeted steps, we are going to battle.
“I think about four hundred gold fredericks must have come into my hands in some five minutes. That was when I should have left, but a strange sensation came over me, a kind of challenge to fate, a sort of desire to give it a fillip, to stick my tongue out at it.” Dostoevsky’s The Gambler was originally titled Roulettenburg—and it was a world he knew well. Wiesbaden, Bad Homburg, Baden-Baden—all the German spas regularly welcomed the intense, red-haired Russian with his increasingly unhappy wife. He would usually arrive filled with confidence, win moderately and briefly, and then dissolve calamitously into loss, guilt, and misery. Loss to Dostoevsky, as to many gamblers, was an aberration: the surprising local failure of an otherwise well-designed system. Its message was not “don’t try to beat the house” but “you’re nearly there; another minor adjustment and it will work perfectly.”
From space the ocean looks like a smooth blue shell; from an airplane, we see an irregular surface like hammered metal; on a raft in a storm, we know only that this wave is upon us and that one is coming—yet we still try to see the pattern of our misfortune and gain advantage from local predictions. The long run reveals the house advantage—but why should we wait for the long run? Life is of the minute; why shouldn’t we seize and exploit departures from the trend?
There are always stories to support the idea of experience as locally wrinkled although globally smooth. William Nelson Darnborough, Bloomington, Illinois’ luckiest if not most famous son, bet on 5 and won on five successive spins of the wheel in 1911. In August 1913, black came up twenty-six times in a row at Monte Carlo; 7 came up six times on wheel 211 at Caesars Palace, Las Vegas, on July 14, 2000. If you had ridden that last particular wave, starting with a $2 chip, reinvesting your winnings and, most important, walking away at the right moment, your fortune would be tidily over $4 billion—were it not for table limits.
No one did ride the wave; in fact, despite the large crowd that gathered and the power of the event on the imagination, the table lost only $300 over those six spins. When black had its reign in Monte Carlo, the casino won millions of francs—because the players were convinced the “law of averages” meant the run had to end sooner than it did—that red somehow ached to redress the balance. The psychology of gambling includes both a conviction that the unusual must happen and a refusal to believe in it when it does.
We are caught by the confusing nature of the long run; just as the imperturbable ocean seen from space will actually combine hurricanes and dead calms, so the same action, repeated over time, can show wide deviations from its normal expected results—deviations that do not themselves break the laws of probability. In fact, they have probabilities of their own.
Take the run of black at Monte Carlo. We will assume that the casino has been kind and lent us a wheel with no zero, so the chance of black on any one spin is 18/36, or exactly half. What is the chance of two black numbers in a row? Well, we can list all the possible outcomes of two spins—black, then black; black, then red; red, then black; red, then red—and conclude that the chance of two black numbers in a row is one in four.
In general, we can describe the results of successive plays as a tree of possibilities, branching at each spin into twice as many potential results: 2 at the first, 4 at the second, 8 at the third. Only one of these possible time-lines will go through black alone, so its chance will be 1/2 on the first spin, 1/4 on the second, 1/8 on the third, and so on. The probability of black coming up twenty-six times in a row, therefore, is 1/2 times itself 26 times, or one chance in around 67 million. That sounds convincingly rare.
Think, though, how many times those wheels have spun. Six tables, with each wheel spun once a minute through the twelve-hour day, 360 days a year—a single salon in Monte Carlo would perform the experiment more than 67 million times roughly every 43 years. If you believed in the law of averages, you might say Monte Carlo has been “due” another such prodigy for al
most half a century.
This notion of “being due”—what is sometimes called the gambler’s fallacy—is a mistake we make because we cannot help it. The problem with life is that we have to live it from the beginning, but it makes sense only when seen from the end. As a result, our whole experience is one of coming to provisional conclusions based on insufficient evidence: reading the signs, gauging the odds. Recent experiments using positron emission tomography (PET) scans have revealed that, even when subjects have been told they are watching a completely random sequence of stimuli, the pattern-finding parts of their brains light up like the Las Vegas strip. We see faces in clouds, hear sermons in stones, find hidden messages in ancient texts. A belief that things reveal meaning through pattern is the gift we brought with us out of Eden.
Our problem, however, is that some things can have shape without structure, the form of meaning without its content. A string of random letters split according to the appropriate word-lengths of English will immediately look like a code. Letters chosen according to an arbitrary rule from a sufficiently long book will spell out messages, thanks to the sheer number of possible combinations: following the fuss about how names of famous rabbis and Israeli politicians had been found this way in the Torah, an Australian mathematician uncovered an equivalent number of famous names—including Indira Gandhi, Abraham Lincoln, and John Kennedy—in Moby-Dick.
It is the same with the numbers generated by roulette: the smoothness of probability in the long term allows any amount of local lumpiness on which to exercise our obsession with pattern. As the sequence of data lengthens, the relative proportions of odd or even, red or black, do indeed approach closer and closer to the 50-50 ratio predicted by probability, but the absolute discrepancy between one and the other will increase. Spin the wheel three times: red comes out twice as often as black, but the difference between them is only one spin. Spin the wheel ten thousand times and the proportion of red to black will be almost exactly even—but as little as a one percent difference between them represents a hundred more spins for one color than another. Those hundred spins may not be distributed evenly through the sequence—as we’ve seen, twenty-six of them might occur in a row. In fact, if you extend your roulette playing indefinitely into the future, you can be certain that one color or another, one number or its neighbor, will at some point occur more or less often than probability suggests—by any margin you choose.
You may object that this is simply a mathematical power play, using the mallet of enormous numbers to pound interesting distinctions into flat uniformity. Yet the statement has meaning at every spin of the wheel and every roll of the dice: it tells us that the game has neither memory nor obligation. It makes no difference whether this is the first ever spin of the wheel or the last of millions. The law of averages rules each play much as the Tsar of All the Russias ruled any one village: absolutely, but at a distance.
Why can’t we believe this? Because we do not observe the law ourselves: the value we attach to things is only loosely related to their innate probability. Although the fixed odds at roulette and the rankings of poker hands do accurately reflect the likelihood of any event, we are not equally interested in them all. The straight flush to which we drew in ’93 at that guy Henry’s house not only outweighs but obliterates the thousands of slushy hands on which we folded in the intervening years. This selective interest, filing away victories and discarding defeats, may well explain why old people consider themselves happier than the young. Our view of life is literary—and we edit well.
Craps, as a probability exercise, shows exactly how much this personal sense of combat with fate diverges from the real odds. The pair of dice differs from the roulette wheel in that the various totals do not have equal chances of coming up. As we have seen, there are 6 × 6 = 36 different ways two dice can make a total. Of these, there’s only one way to make 2 or 12; two ways for 3 or 11; three ways for 4 and 10; four ways for 5 and 9; five ways for 6 or 8; and six ways for 7.
The game and its payoffs are designed around these probabilities: at the first roll, you can win with the most common throw (7; you can also win with 11), and lose with the rare ones (2, 12, 3). Once you are in the game, though, the position is reversed: having established your “point” (which can be 4, 5, 6, 8, 9, or 10), you need to roll it again before rolling the most common total, 7. Some points are easier to make than others. Trying to roll another 4 or 10 is tough; if you’re the shooter, you will find the crowd eagerly plunking down chips on the “Don’t Pass” box. 6 and 8 are more likely; you’ll have the room rooting for you.
Is there an overall probability to craps? Is it a fair proposition to pick up the dice and give them a speculative toss? The way to calculate this is to extend the idea of cases or events: just as we can add together the number of ways you can make 7 and compare that with the total number of cases, so we can add together the number of ways you can make your point before crapping out for each point, and compare it with the total number of possible outcomes.
Why, you may wonder, are we allowed to add all the probabilities together? Because they do not overlap; they are mutually exclusive. The parallel universe in which you won by shooting a natural first throw is not the universe in which you won by making a point of 10 the hard way. Here, we can total everything that might happen and use it to measure the one thing that does.
So, adding up all the probabilities of victory in craps, you find that the shooter should prevail a total of a little over 49 percent of the time, which means there is less chance of a win for the guy bellying up to the dice table than there is for the grandmother slipping her plaquette onto black at Monte Carlo. The shooter beats the even-money roulette player in Las Vegas, however, because American wheels have a double zero as well as a zero, thereby doubling the house advantage. Why, therefore, is it so desirable to be the shooter? Perhaps it’s a chance to act out the combat with destiny, pitting our gestures and rituals against the force of randomness, blowing our spirit power into the unresponsive cellulose acetate.
A pair of dice can only tell you 36 things—some more welcome than others. A pack of cards, though, rapidly expands the universe of possibilities, giving the gambler some sense of measureless immensity. If you start with any one of the fifty-two cards, there are fifty-one possible candidates for the next card; for each of them, fifty choices for the third—and so on. The number of possibilities for the whole deck (52 × 51 × 50 × ... × 3 × 2 × 1; called “fifty-two factorial” and written, appropriately, 52!) is a number 68 digits long—far greater than the number of grains of sand on every beach in the world, or of molecules of water in its oceans. In fact, assuming it is well shuffled, there is no probabilistic reason that any one pack of cards should be in the same order as any other that has ever been dealt in the entire history of card playing.
Being “well shuffled” is actually a more complex matter than the simple phrase suggests. You might assume that shuffling a deck of cards is much the same as spinning the roulette wheel or rolling the dice: a straightforward way to invite the spirit of chance into play, leaving the table clear for your combat with fate or the opposing gambler. Shuffling, though, does not start at the same point each time. Spin the wheel, then spin the wheel again—the two events are independent: red or 35 is equally likely to come up on either spin. A deck of cards, though, does not reset to zero between shuffles: if you cut and riffle or dovetail once, then twice, you have taken two distinct steps away from the original order of the deck. The result of the second shuffle depends on the first; if you somehow managed to perform that second shuffle on the original deck, the result would be different.
This kind of sequential arbitrariness is called a random walk. Let’s say you went to the park with a small child, a tennis ball, and a dog, and sensibly set out the picnic blanket in the middle of a large open space. The child (let’s call her Lucy) and dog (Pupkin), fizzing with energy, set off with the ball. Lucy can throw the ball ten feet each time but has almost no control over its direct
ion. Pupkin doesn’t yet understand retrieval: he usually runs to the ball, lies down on it, and barks until Lucy comes to take it from him. When you look up, how far will Lucy be from the blanket? As you can guess, the distance depends on how many times she has thrown the ball; even though the direction of each throw is random, the process is sequential: she couldn’t be more than thirty feet away after three throws, although it’s possible she could be back on the blanket, skipping through the pie, after five. Indeed, it is certain that a two-dimensional random walk like this will at some time pass again through its starting point. Given how far she can throw, you can gauge the likelihood of Lucy’s being a given distance from your blanket as a function of the number of throws: for n throws, the average final distance over several picnics’ worth of random walks will be √n times the length of one throw.
These so-called stochastic processes show up everywhere randomness is applied to the output of another random function. They provide, for instance, a method for describing the chance component of financial markets: not every value of the Dow is possible every day; the range of chance fluctuation centers on the opening price. Similarly, shuffling takes the output of the previous shuffle as its input. So, if you’re handed a deck in a given order, how much shuffling does it need to be truly random?
Persi Diaconis is one of the great names in modern probability; before he became a professor of mathematics and statistics at Stanford, he was a professional magician. He knows how far from random a deck can be and how useful that is for creating improbable effects—the reason why magicians seem so often to be nervously shuffling the deck as they do their patter.