Chapter 5: The Dueling Laws Of Large and Small Numbers

[amk]

I came across Chapter 5 of the book "A Drunkards Walk/ How Randomenss Rules Our Lives".

I found it very interesting. Hope we can have a good discussion about it. I still have to read the entire book.  Can be found at :

Chapter 5: The Dueling Laws of Large and Small Numbers

Here is the excerpt: Page 81
                                                             
                                                                        CHAPTER 5


In their work, Cardano, Galileo, and Pascal assumed thatthe probabilities relevant to the problems they tackled wereknown. Galileo, for example, assumed that a die has an equalchance of landing on any of its six faces. But how solid is such“knowledge”? The grand duke’s dice were probably designed not tofavor any face, but that doesn’t mean fairness was actually achieved.Galileo could have tested his assumption by observing a number of tosses and recording how often each face came up. If he had repeatedthe test several times, however, he would probably have found aslightly different distribution each time, and even small deviationsmight have mattered, given the tiny differential he was asked toexplain. In order to make the early work on randomness applicable tothe real world, that issue had to be addressed: What is the connectionbetween underlying probabilities and observed results? What does itmean, from a practical point of view, when we say the chances are1 in 6 a die will land on 2? If it doesn’t mean that in any series of tosses the die will land on the 2 exactly 1 time in 6, then on what dowe base our belief that the chances of throwing a 2 really are 1 in 6?And what does it mean when a doctor says that a drug is 70 percenteffective or has serious side effects in 1 percent of the cases or when apoll finds that a candidate has support of 36 percent of voters? These are deep questions, related to the very meaning of the concept of ran-domness, a concept mathematicians still like to debate.I recently engaged in such a discussion one warm spring day witha statistician visiting from Hebrew University, Moshe, who sat acrossthe lunch table from me at Caltech. Between spoonfuls of nonfatyogurt, Moshe espoused the opinion that truly random numbers donot exist. “There is no such thing,” he said. “Oh, they publish chartsand write computer programs, but they are just fooling themselves.No one has ever found a method of producing randomness that’s anybetter than throwing a die, and throwing a die just won’t do it.”Moshe waved his white plastic spoon at me. He was agitated now.I felt a connection between his feelings about randomness and hisreligious convictions. Moshe is an Orthodox Jew, and I know thatmany religious people have problems thinking God can allow ran-domness to exist. “Suppose you want a string of
N
random numbersbetween 1 and 6,” he told me. “You throw a die
N
times and recordthe string of
N
numbers that comes up. Is that a random string?”No, he claimed, because no one can make a perfect die. Therewill always be some faces that are favored and some that are disfa-vored. It might take 1,000 throws to notice the difference, or 1 bil-lion, but eventually you will notice it. You’ll see more 4s than 6s ormaybe fewer. Any artificial device is bound to suffer from that ï¬,aw,he said, because human beings do not have access to perfection.That may be, but Nature does, and truly random events do occur onthe atomic level. In fact, that is the very basis of quantum theory, andso we spent the rest of our lunch in a discussion of quantum optics.Today cutting-edge quantum generators produce truly randomnumbers from the toss of Nature’s perfect quantum dice. In thepast the perfection necessary for randomness was indeed an elusivegoal. One of the most creative approaches came from New YorkCity’s Harlem crime syndicates around 1920.

Needing a daily supplyof five-digit random numbers for an illegal lottery, the racketeersthumbed their noses at the authorities by employing the last five dig-its of the U.S. Treasury balance. (At this writing the U.S. governmentis in debt by $8,995,800,515,946.50, or $29,679.02 per person, so

today the racketeers could have obtained their five digits from the percapita debt!) Their so-called Treasury lottery ran afoul of not onlycriminal law, however, but also scientific law, for according to a rulecalled Benford’s law, numbers arising in this cumulative fashion arenot random but rather are biased in favor of the lower digits.Benford’s law was discovered not by a fellow named Benford butby the American astronomer Simon Newcomb. Around 1881, New-comb noticed that the pages of books of logarithms that dealt withnumbers beginning with the numeral 1 were dirtier and more frayedthan the pages corresponding to numbers beginning with thenumeral 2, and so on, down to the numeral 9, whose pages, in com-parison, looked clean and new. Assuming that in the long run, wearwas proportional to amount of use, Newcomb concluded from hisobservations that the scientists with whom he shared the book wereworking with data that reï¬,ected that distribution of digits. The law’scurrent name arose after Frank Benford noticed the same thing, in1938, when scrutinizing the log tables at the General ElectricResearch Laboratory in Schenectady, New York. But neither manproved the law. That didn’t happen until 1995, in work by Ted Hill, amathematician at the Georgia Institute of Technology.According to Benford’s law, rather than all nine digits’ appearingwith equal frequency, the number 1 should appear as the first digit indata about 30 percent of the time; the digit 2, about 18 percent of thetime; and so on, down to the digit 9, which should appear as the firstdigit about 5 percent of the time. A similar law, though less pro-nounced, applies to later digits. Many types of data obey Benford’slaw, in particular, financial data. In fact, the law seems tailor-madefor mining large amounts of financial data in search of fraud.One famous application involved a young entrepreneur namedKevin Lawrence, who raised $91 million to create a chain of high-tech health clubs. Engorged with cash, Lawrence raced into action,hiring a bevy of executives and spending his investors’ money asquickly as he had raised it. That would have been fine except for onedetail: he and his cohorts were spending most of the money not onthe business but on personal items. And since several homes, twenty personal watercraft, forty-seven cars (including five Hummers, fourFerraris, three Dodge Vipers, two DeTomaso Panteras, and a Lam-borghini Diablo), two Rolex watches, a twenty-one-carat diamondbracelet, a $200,000 samurai sword, and a commercial-grade cottoncandy machine would have been difficult to explain as necessarybusiness expenditures, Lawrence and his pals tried to cover theirtracks by moving investors’ money through a complex web of bankaccounts and shell companies to give the appearance of a bustlingand growing business. Unfortunately for them, a suspicious forensicaccountant named Darrell Dorrell compiled a list of over 70,000numbers representing their various checks and wire transfers andcompared the distribution of digits with Benford’s law. The numbersfailed the test.

That, of course, was only the beginning of the investi-gation, but from there the saga unfolded predictably, ending the daybefore Thanksgiving 2003, when, ï¬,anked by his attorneys and clad inlight blue prison garb, Kevin Lawrence was sentenced to twenty yearswithout possibility of parole. The IRS has also studied Benford’s lawas a way to identify tax cheats. One researcher even applied the law tothirteen years of Bill Clinton’s tax returns. They passed the test.

Presumably neither the Harlem syndicate nor its customersnoticed these regularities in their lottery numbers. But had peoplelike Newcomb, Benford, or Hill played their lottery, in principle theycould have used Benford’s law to make favorable bets, earning a nicesupplement to their scholar’s salary.In 1947, scientists at the Rand Corporation needed a large tableof random digits for a more admirable purpose: to help find approxi-mate solutions to certain mathematical equations employing a tech-nique aptly named the Monte Carlo method. To generate the digits,they employed electronically generated noise, a kind of electronicroulette wheel. Is electronic noise random? That is a question as sub-tle as the definition of randomness itself.In 1896 the American philosopher Charles scammer Sanders Peirce wrotethat a random sample is one “taken according to a precept or methodwhich, being applied over and over again indefinitely, would in thelong run result in the drawing of any one of a set of instances as often as any other set of the same number.”

That is called the frequencyinterpretation of randomness. The main alternative to it is called thesubjective interpretation. Whereas in the frequency interpretationyou judge a sample by the way it turned out, in the subjective inter-pretation you judge a sample by the way it is produced. According tothe subjective interpretation, a number or set of numbers is consid-ered random if we either don’t know or cannot predict how theprocess that produces it will turn out.The difference between the two interpretations is more nuancedthan it may seem. For example, in a perfect world a throw of a diewould be random by the first definition but not by the second, sinceall faces would be equally probable but we could (in a perfect world)employ our exact knowledge of the physical conditions and the lawsof physics to determine before each throw exactly how the die willland. In the imperfect real world, however, a throw of a die is randomaccording to the second definition but not the first. That’s because, asMoshe pointed out, owing to its imperfections, a die will not land oneach face with equal frequency; nevertheless, because of our limita-tions we have no prior knowledge about any face being favored overany other.In order to decide whether their table was random, the Rand sci-entists subjected it to various tests. Upon closer inspection, their sys-tem was shown to have biases, just like Moshe’s archetypallyimperfect dice.

The Rand scientists made some refinements to theirsystem but never managed to completely banish the regularities. AsMoshe said, complete chaos is ironically a kind of perfection. Still,the Rand numbers proved random enough to be useful, and the com-pany published them in 1955 under the catchy title A Million Ran-dom Digits.
In their research the Rand scientists ran into a roulette-wheelproblem that had been discovered, in some abstract way, almost acentury earlier by an Englishman named Joseph Jagger.

Jagger wasan engineer and a mechanic in a cotton factory in Yorkshire, and sohe had an intuitive feel for the capabilitiesâ€"and the shortcomingsâ€"of machinery and one day in 1873 turned his intuition and fertile mind from cotton to cash. How perfectly, he wondered, can theroulette wheels in Monte Carlo really work?The roulette wheelâ€"invented, at least according to legend, byBlaise Pascal as he was tinkering with an idea for a perpetual-motionmachineâ€"is basically a large bowl with partitions (called frets) thatare shaped like thin slices of pie. When the wheel is spun, a marblefirst bounces along the rim of the bowl but eventually comes to restin one of the compartments, which are numbered 1 through 36, plus0 (and 00 on American roulette wheels). The bettor’s job is simple: toguess in which compartment the marble will land. The existence of roulette wheels is pretty good evidence that legitimate psychics don’texist, for in Monte Carlo if you bet $1 on a compartment and themarble lands there, the house pays you $35 (plus your initial dollar).If psychics really existed, you’d see them in places like that, hootingand dancing and pushing wheelbarrows of cash down the street, andnot on Web sites calling themselves Zelda Who Knows All and SeesAll and offering twenty-four-hour free online love advice in competi-tion with about 1.2 million other Web psychics (according toGoogle). For me both the future and, increasingly, the past unfortu-nately appear obscured by a thick fog. But I do know one thing: mychances of losing at European roulette are 36 out of 37; my chancesof winning, 1 out of 37. That means that for every $1 I bet, the casinostands to win about  2.7¢. Depending on my state of mind, it’s either the price Ipay for the enjoyment of watching a little marble bounce around abig shiny wheel or else the price I pay for the opportunity of havinglightning strike me (in a good way). At least that is how it is supposedto work.But does it? Only if the roulette wheels are perfectly balanced,thought Jagger, and he had worked with enough machines to shareMoshe’s point of view. He was willing to bet they weren’t. So he gath-ered his savings, traveled to Monte Carlo, and hired six assistants,one for each of the casino’s six roulette wheels. Every day his assis-tants observed the wheels, writing down every number that came upin the twelve hours the casino was open. Every night, back in his
hotel room, Jagger analyzed the numbers. After six days, he had notdetected any bias in five of the wheels, but on the sixth wheel ninenumbers came up noticeably more often than the others. And so onthe seventh day he headed to the casino and started to bet heavily onthe nine favored numbers: 7, 8, 9, 17, 18, 19, 22, 28, and 29.When the casino shut that night, Jagger was up $70,000. His win-nings did not go without notice. Other patrons swarmed his table,tossing down their own cash to get in on a good thing. And casinoinspectors were all over him, trying to decipher his system or, better,catch him cheating. By the fourth day of betting, Jagger had amassed$300,000, and the casino’s managers were desperate to get rid of themystery guy, or at least thwart his scheme. One imagines this beingaccomplished by a burly fellow from Brooklyn. Actually the casinoemployees did something far more clever.On the fifth day, Jagger began to lose. His losing, like his winning,was not something you could spot immediately. Both before and afterthe casino’s trick, he would win some and lose some, only now he lostmore often than he won instead of the other way around. With thecasino’s small margin, it would take some pretty diligent betting todrain Jagger’s funds, but after four days of sucking in casino money,he wasn’t about to let up on the straw. By the time his change of luckdeterred him, Jagger had lost half his fortune. One may imagine thatby then his moodâ€"not to mention the mood of his hangers-onâ€"wassour. How could his scheme have suddenly failed?Jagger at last made an astute observation. In the dozens of hourshe had spent winning, he had come to notice a tiny scratch on theroulette wheel. This scratch was now absent. Had the casino kindlytouched it up so that he could drive them to bankruptcy in style? Jag-ger guessed not and checked the other roulette wheels. One of themhad a scratch. The casino managers had correctly guessed that Jag-ger’s days of success were somehow related to the wheel he was play-ing, and so overnight they had switched wheels. Jagger relocated andagain began to win. Soon he had pumped his winnings past wherethey had been, to almost half a million.Unfortunately for Jagger, the casino’s managers, finally zeroing in
on his scheme, found a new way to thwart him. They decided tomove the frets each night after closing, turning them along the wheelso that each day the wheel’s imbalance would favor different num-bers, numbers unknown to Jagger. Jagger started losing again andfinally quit. His gambling career over, he left Monte Carlo with$325,000 in hand, about $5 million in today’s dollars. Back home, heleft his job at the mill and invested his money in real estate.It may appear that Jagger’s scheme had been a sure thing, but itwasn’t. For even a perfectly balanced wheel will not come up on 0, 1,2, 3, and so on, with exactly equal frequencies, as if the numbers inthe lead would politely wait for the laggards to catch up. Instead,some numbers are bound to come up more often than average andothers less often. And so even after six days of observations, thereremained a chance that Jagger was wrong. The higher frequencieshe observed for certain numbers may have arisen by chance and maynot have reï¬,ected higher probabilities. That means that Jagger, too,had to face the question we raised at the start of this chapter: given aset of underlying probabilities, how closely can you expect yourobservations of a system to conform to those probabilities? Just as Pas-cal’s work was done in the new climate of (the scientific) revolution,so this question would be answered in the midst of a revolution, thisone in mathematicsâ€"the invention of calculus.