Drunkard's Walk - Leonard Mlodinow [49]
Zeno’s paradox concerns the amount of time it takes to make the journey, not the distance covered. If the student were forced to take individual steps to cover each of Zeno’s intervals, she would indeed be in some time trouble (not to mention her having to overcome the difficulty of taking submillimeter steps)! But if she is allowed to move at constant speed without pausing at Zeno’s imaginary checkpoints—and why not?—then the time it takes to travel each of Zeno’s intervals is proportional to the distance covered in that interval, and so since the total distance is finite, as is the total time—and fortunately for all of us—motion is possible after all.
Though the modern concept of limits wasn’t worked out until long after Zeno’s life, and even Bernoulli’s—it came in the nineteenth century14—it is this concept that informs the spirit of calculus, and it is in this spirit that Jakob Bernoulli attacked the relationship between probabilities and observation. In particular, Bernoulli investigated what happens in the limit of an arbitrarily large number of repeated observations. Toss a (balanced) coin 10 times and you might observe 7 heads, but toss it 1 zillion times and you’ll most likely get very near 50 percent. In the 1940s a South African mathematician named John Kerrich decided to test this out in a practical experiment, tossing a coin what must have seemed like 1 zillion times—actually it was 10,000—and recording the results of each toss.15 You might think Kerrich would have had better things to do, but he was a war prisoner at the time, having had the bad luck of being a visitor in Copenhagen when the Germans invaded Denmark in April 1940. According to Kerrich’s data, after 100 throws he had only 44 percent heads, but by the time he reached 10,000, the number was much closer to half: 50.67 percent. How do you quantify this phenomenon? The answer to that question was Bernoulli’s accomplishment.
According to the historian and philosopher of science Ian Hacking, Bernoulli’s work “came before the public with a brilliant portent of all the things we know about it now; its mathematical profundity, its unbounded practical applications, its squirming duality and its constant invitation for philosophizing. Probability had fully emerged.” In Bernoulli’s more modest words, his study proved to be one of “novelty, as well as…high utility.” It was also an effort, Bernoulli wrote, of “grave difficulty.”16 He worked on it for twenty years.
JAKOB BERNOULLI called the high point of his twenty-year effort his “golden theorem.” Modern versions of it that differ in their technical nuance go by various names: Bernoulli’s theorem, the law of large numbers, and the weak law of large numbers. The phrase law of large numbers is employed because, as we’ve said, Bernoulli’s theorem concerns the way results reflect underlying probabilities when we make a large number of observations. But we’ll stick with Bernoulli’s terminology and call his theorem the golden theorem because we will be discussing it in its original form.17
Although Bernoulli’s interest lay in real-world applications, some of his favorite examples involved an item not found in most households: an urn filled with colored pebbles. In one scenario, he envisioned the urn holding 3,000 white pebbles and 2,000 black ones, a ratio of 60 percent white to 40 percent black. In this example you conduct a series of blind drawings from the urn “with replacement”—that is, replacing each pebble before drawing the next in order not to alter the 3:2 ratio. The a priori chances of drawing a white pebble are then 3 out of 5, or 60 percent, and so in this example Bernoulli