Drunkard's Walk - Leonard Mlodinow [43]
According to Benford’s law, rather than all nine digits’ appearing with equal frequency, the number 1 should appear as the first digit in data about 30 percent of the time; the digit 2, about 18 percent of the time; and so on, down to the digit 9, which should appear as the first digit about 5 percent of the time. A similar law, though less pronounced, applies to later digits. Many types of data obey Benford’s law, in particular, financial data. In fact, the law seems tailor-made for mining large amounts of financial data in search of fraud.
One famous application involved a young entrepreneur named Kevin Lawrence, who raised $91 million to create a chain of high-tech health clubs.2 Engorged with cash, Lawrence raced into action, hiring a bevy of executives and spending his investors’ money as quickly as he had raised it. That would have been fine except for one detail: he and his cohorts were spending most of the money not on the business but on personal items. And since several homes, twenty personal watercraft, forty-seven cars (including five Hummers, four Ferraris, three Dodge Vipers, two DeTomaso Panteras, and a Lamborghini Diablo), two Rolex watches, a twenty-one-carat diamond bracelet, a $200,000 samurai sword, and a commercial-grade cotton candy machine would have been difficult to explain as necessary business expenditures, Lawrence and his pals tried to cover their tracks by moving investors’ money through a complex web of bank accounts and shell companies to give the appearance of a bustling and growing business. Unfortunately for them, a suspicious forensic accountant named Darrell Dorrell compiled a list of over 70,000 numbers representing their various checks and wire transfers and compared the distribution of digits with Benford’s law. The numbers failed the test.3 That, of course, was only the beginning of the investigation, but from there the saga unfolded predictably, ending the day before Thanksgiving 2003, when, flanked by his attorneys and clad in light blue prison garb, Kevin Lawrence was sentenced to twenty years without possibility of parole. The IRS has also studied Benford’s law as a way to identify tax cheats. One researcher even applied the law to thirteen years of Bill Clinton’s tax returns. They passed the test.4
Presumably neither the Harlem syndicate nor its customers noticed these regularities in their lottery numbers. But had people like Newcomb, Benford, or Hill played their lottery, in principle they could have used Benford’s law to make favorable bets, earning a nice supplement to their scholar’s salary.
In 1947, scientists at the Rand Corporation needed a large table of random digits for a more admirable purpose: to help find approximate solutions to certain mathematical equations employing a technique aptly named the Monte Carlo method. To generate the digits, they employed electronically generated noise, a kind of electronic roulette wheel. Is electronic noise random? That is a question as subtle as the definition of randomness itself.
In 1896 the American philosopher Charles Sanders Peirce wrote that a random sample is one “taken according to a precept or method which, being applied over and over again indefinitely, would in the long run result in the drawing of any one of a set of instances as often as any other set of the same number.”5 That is called the frequency interpretation of randomness. The main alternative to it is called the subjective interpretation. Whereas in the frequency interpretation you judge a sample by the way it turned out, in the subjective interpretation you judge a sample