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of tosses and recording how often each face came up. If he had repeated the test several times, however, he would probably have found a slightly different distribution each time, and even small deviations might have mattered, given the tiny differential he was asked to explain. In order to make the early work on randomness applicable to the real world, that issue had to be addressed: What is the connection between underlying probabilities and observed results? What does it mean, from a practical point of view, when we say the chances are 1 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 do we 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 percent effective or has serious side effects in 1 percent of the cases or when a poll finds that a candidate has support of 36 percent of voters? These are deep questions, related to the very meaning of the concept of randomness, a concept mathematicians still like to debate.

I recently engaged in such a discussion one warm spring day with a statistician visiting from Hebrew University, Moshe, who sat across the lunch table from me at Caltech. Between spoonfuls of nonfat yogurt, Moshe espoused the opinion that truly random numbers do not exist. “There is no such thing,” he said. “Oh, they publish charts and write computer programs, but they are just fooling themselves. No one has ever found a method of producing randomness that’s any better 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 his religious convictions. Moshe is an Orthodox Jew, and I know that many religious people have problems thinking God can allow randomness to exist. “Suppose you want a string of N random numbers between 1 and 6,” he told me. “You throw a die N times and record the string of N numbers that comes up. Is that a random string?”

No, he claimed, because no one can make a perfect die. There will always be some faces that are favored and some that are disfavored. It might take 1,000 throws to notice the difference, or 1 billion, but eventually you will notice it. You’ll see more 4s than 6s or maybe fewer. Any artificial device is bound to suffer from that flaw, he said, because human beings do not have access to perfection. That may be, but Nature does, and truly random events do occur on the atomic level. In fact, that is the very basis of quantum theory, and so we spent the rest of our lunch in a discussion of quantum optics.

Today cutting-edge quantum generators produce truly random numbers from the toss of Nature’s perfect quantum dice. In the past the perfection necessary for randomness was indeed an elusive goal. One of the most creative approaches came from New York City’s Harlem crime syndicates around 1920.1 Needing a daily supply of five-digit random numbers for an illegal lottery, the racketeers thumbed their noses at the authorities by employing the last five digits of the U.S. Treasury balance. (At this writing the U.S. government is 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 per capita debt!) Their so-called Treasury lottery ran afoul of not only criminal law, however, but also scientific law, for according to a rule called Benford’s law, numbers arising in this cumulative fashion are not random but rather are biased in favor of the lower digits.

Benford’s law was discovered not by a fellow named Benford but by the American astronomer Simon Newcomb. Around 1881, Newcomb noticed that the pages of books of logarithms that dealt with numbers beginning with the numeral 1 were dirtier and more frayed than the pages corresponding to numbers beginning with the numeral 2, and so on, down to the numeral 9, whose pages, in comparison, looked clean and new. Assuming that in the long run, wear was proportional to amount