Drunkard's Walk - Leonard Mlodinow [26]
Cardano’s Book on Games of Chance covers card games, dice, backgammon, and astragali. It is not perfect. In its pages are reflected Cardano’s character, his crazy ideas, his wild temper, the passion with which he approached every undertaking—and the turbulence of his life and times. It considers only processes—such as the toss of a die or the dealing of a playing card—in which one outcome is as likely as another. And some points Cardano gets wrong. Still, The Book on Games of Chance represents a beachhead, the first success in the human quest to understand the nature of uncertainty. And Cardano’s method of attacking questions of chance is startling both in its power and in its simplicity.
Not all the chapters of Cardano’s book treat technical issues. For instance, chapter 26 is titled “Do Those Who Teach Well Also Play Well?” (he concludes, “It seems to be a different thing to know and to execute”). Chapter 29 is called “On the Character of Players” (“There are some who with many words drive both themselves and others from their proper senses”). These seem more “Dear Abby” than “Ask Marilyn.” But then there is chapter 14, “On Combined Points” (on possibilities). There Cardano states what he calls “a general rule”—our law of the sample space.
The term sample space refers to the idea that the possible outcomes of a random process can be thought of as the points in a space. In simple cases the space might consist of just a few points, but in more complex situations it can be a continuum, just like the space we live in. Cardano didn’t call it a space, however: the notion that a set of numbers could form a space was a century off, awaiting the genius of Descartes, his invention of coordinates, and his unification of algebra and geometry.
In modern language, Cardano’s rule reads like this: Suppose a random process has many equally likely outcomes, some favorable (that is, winning), some unfavorable (losing). Then the probability of obtaining a favorable outcome is equal to the proportion of outcomes that are favorable. The set of all possible outcomes is called the sample space. In other words, if a die can land on any of six sides, those six outcomes form the sample space, and if you place a bet on, say, two of them, your chances of winning are 2 in 6.
A word on the assumption that all the outcomes are equally likely. Obviously that’s not always true. The sample space for observing Oprah Winfrey’s adult weight runs (historically) from 145 pounds to 237 pounds, and over time not all weight intervals have proved equally likely.12 The complication that different possibilities have different probabilities can be accounted for by associating the proper odds with each possible outcome—that is, by careful accounting. But for now we’ll look at examples in which all outcomes are equally probable, like those Cardano analyzed.
The potency of Cardano’s rule goes hand in hand with certain subtleties. One lies in the meaning of the term outcomes. As late as the eighteenth century the famous French mathematician Jean Le Rond d’Alembert, author of several works on probability, misused the concept when he analyzed the toss of two coins.13 The number of heads that turns up in those two tosses can be 0, 1, or 2. Since there are three outcomes, Alembert reasoned, the chances of each must be 1 in 3. But Alembert was mistaken.
One of the greatest deficiencies of Cardano’s work was that he made no systematic analysis of the different ways in which a series of events, such as coin tosses, can turn out. As we shall see in the next chapter, no one did that until the following century. Still,