Drunkard's Walk - Leonard Mlodinow [50]
The urn example is a good one because the same mathematics that describes drawing pebbles from an urn can be employed to describe any series of trials in which each trial has two possible outcomes, as long as those outcomes are random and the trials are independent of each other. Today such trials are called Bernoulli trials, and a series of Bernoulli trials is a Bernoulli process. When a random trial has two possible outcomes, one is often arbitrarily labeled “success” and the other “failure.” The labeling is not meant to be literal and sometimes has nothing to do with the everyday meaning of the words—say, in the sense that if you can’t wait to read on, this book is a success, and if you are using this book to keep yourself and your sweetheart warm after the logs burned down, it is a failure. Flipping a coin, deciding to vote for candidate A or candidate B, giving birth to a boy or girl, buying or not buying a product, being cured or not being cured, even dying or living are examples of Bernoulli trials. Actions that have multiple outcomes can also be modeled as Bernoulli trials if the question you are asking can be phrased in a way that has a yes or no answer, such as “Did the die land on the number 4?” or “Is there any ice left on the North Pole?” And so, although Bernoulli wrote about pebbles and urns, all his examples apply equally to these and many other analogous situations.
With that understanding we return to the urn, 60 percent of whose pebbles are white. If you draw 100 pebbles from the urn (with replacement), you might find that exactly 60 of them are white, but you might also draw just 50 white pebbles or 59. What are the chances that you will draw between 58 percent and 62 percent white pebbles? What are the chances you’ll draw between 59 percent and 61 percent? How much more confident can you be if instead of 100, you draw 1,000 pebbles or 1 million? You can never be 100 percent certain, but can you draw enough pebbles to make the chances 99.9999 percent certain that you will draw, say, between 59.9 percent and 60.1 percent white pebbles? Bernoulli’s golden theorem addresses questions such as these.
In order to apply the golden theorem, you must make two choices. First, you must specify your tolerance of error. How near to the underlying proportion of 60 percent are you demanding that your series of trials come? You must choose an interval, such as plus or minus 1 percent or 2 percent or 0.00001 percent. Second, you must specify your tolerance of uncertainty. You can never be 100 percent sure a trial will yield the result you are aiming for, but you can ensure that you will get a satisfactory result 99 times out of 100 or 999 out of 1,000.
The golden theorem tells you that it is always possible to draw enough pebbles to be almost certain that the percentage of white pebbles you draw will be near 60 percent no matter how demanding you want to be in your personal definition of almost certain and near. It also gives a numerical formula for calculating the number of trials that are “enough,” given those definitions.
The first part of the law was a conceptual triumph, and it is the only part that survives in modern versions of the theorem. Concerning the second part—Bernoulli’s formula—it is important to understand that although the golden theorem specifies a number of trials that is sufficient to meet your goals of confidence and accuracy, it does not say you can’t accomplish those goals with fewer trials. That doesn’t affect the first part of the theorem, for which it is enough to know simply that the number of trials specified is finite. But Bernoulli also intended the number given by his formula to be of practical use. Unfortunately, in most practical applications it isn’t. For instance, here is a numerical example Bernoulli worked out himself, although I have changed the context: Suppose 60 percent of the voters in Basel support the mayor. How many