Against the Gods: The Remarkable Story of Risk
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This first concept of probability is much the older of the two; the idea of measuring probability emerged much later. This older sense developed over time from the idea of approbation: how much can we accept of what we know? In Galileo's context, probability was how much we could approve of what we were told. In Leibniz's more modern usage, it was how much credibility we could give the evidence.
The more recent view did not emerge until mathematicians had developed a theoretical understanding of the frequencies of past events. Cardano may have been the first to introduce the statistical side of the theory of probability, but the contemporary meaning of the word during his lifetime still related only to the gut side and had no connection with what he was trying to accomplish in the way of measurement.
Cardano had a sense that he was onto something big. He wrote in his autobiography that Liber de Ludo Aleae was among his greatest achievements, claiming that he had "discovered the reason for a thousand astounding facts." Note the words "reason for." The facts in the book about the frequency of outcomes were known to any gambler; the theory that explains such frequencies was not. In the book, Cardano issues the theoretician's customary lament: ". . . these facts contribute a great deal to understanding but hardly anything to practical play."
In his autobiography Cardano says that he wrote Liber de Ludo Aleae in 1525, when he was still a young man, and rewrote it in 1565. Despite its extraordinary originality, in many ways the book is a mess. Cardano put it together from rough notes, and solutions to problems that appear in one place are followed by solutions that employ entirely different methods in another place. The lack of any systematic use of mathematical symbols complicates matters further. The work was never published during Cardano's lifetime but was found among his manuscripts when he died; it was first published in Basle in 1663. By that time impressive progress in the theory of probability had been made by others who were unaware of Cardano's pathfinding efforts.
Had a century not passed before Cardano's work became available for other mathematicians to build on, his generalizations about probabilities in gambling would have significantly accelerated the advance of mathematics and probability theory. He defined, for the first time, what is now the conventional format for expressing probability as a fraction: the number of favorable outcomes divided by the "circuit"-that is, the total number of possible outcomes. For example, we say the chance of throwing heads is 50/50, heads being one of two equally likely cases. The probability of drawing a queen from a full deck of cards is 1/13, as there are four queens in a deck of 52 cards; the chance of drawing the queen of spades, however, is 1/52, for the deck holds only one queen of spades.
Let us follow Cardano's line of reasoning as he details the probability of each throw in a game of dice.* In the following paragraph from Chapter 15 of Liber de Ludo Aleae, "On the cast of one die," he is articulating general principles that no one had ever set forth before:
One-half the total number of faces always represents equality; thus the chances are equal that a given point will turn up in three throws, for the total circuit is completed in six, or again that one of three given points will turn up in one throw. For example, I can as easily throw one, three or five as two, four or six. The wagers there are laid in accordance with this equality if the die is honest.14
In carrying this line of argument forward, Cardano calculates the probability of throwing any of two numbers-say, either a 1 or a 2on a single throw. The answer is one chance out of three, or 33%, because the problem involves two numbers out of a "circuit" of six faces on the die. He also calculates the probability of repeating favorable throws with a single die. The probability of throwing a 1 or a 2 twice in succession is 1/9, which is the square of one chance out of three, or 1 /3 multiplied by itself The probability of throwing a 1 or a 2 three times in succession would be 1/27, or 1/3 x 1/3 x 1/3, while the probability of throwing a 1 or a 2 four times in succession would be 1/3 to the fourth power.
Cardano goes on to figure the probability of throwing a 1 or a 2 with a pair of dice, instead of with a single die. If the probability of throwing a 1 or a 2 with a single die is one out of three, intuition would suggest that throwing a 1 or a 2 with two dice would be twice as great, or 67%. The correct answer is actually five out of nine, or 55.6%. When throwing two dice, there is one chance out of nine that a 1 or a 2 will come up on both dice on the same throw, but the probability of a 1 or a 2 on either die has already been accounted for; hence, we must deduct that one-ninth probability from the 67% that intuition predicts. Thus, 1/3 + 1/3 - 1/9 = 5/9.
Cardano builds up to games for more dice and more wins more times in succession. Ultimately, his research leads him to generalizations about the laws of chance that convert experimentation into theory.
Cardano took a critical step in his analysis of what happens when we shift from one die to two. Let us walk again through his line of reasoning, but in more detail. Although two dice will have a total of twelve sides, Cardano does not define the probability of throwing a 1 or a 2 with two dice as being limited to only twelve possible outcomes. He recognized that a player might, for example, throw a 3 on one die and a 4 on the other die, but that the player could equally well throw a 4 on the first die and a 3 on the second.
The number of possible combinations that make up the "circuit"the total number of possible outcomes-adds up to a lot more than the total number of twelve faces found on the two dice. Cardano's recognition of the powerful role of combinations of numbers was the most important step he took in developing the laws of probability.
The game of craps provides a useful illustration of the importance of combinations in figuring probabilities. As Cardano demonstrated, throwing a pair of six-sided dice will produce, not eleven (from two to twelve), but thirty-six possible combinations, all the way from snake eyes (two ones) to box cars (double six).
Seven, the key number in craps, is the easiest to throw. It is six times as likely as double-one or double-six and three times as likely as eleven, the other key number. The six different ways to arrive at seven are 6 + 1, 5 + 2, 4 + 3, 3 + 4, 2 + 5, and 1 + 6; note that this pattern is nothing more than the sums of each of three different combinations-5 and 2, 4 and 3, and 1 and 6. Eleven can show up only two ways, because it is the sum of only one combination: 5 + 6 or 6 + 5. There is only one way for each of double-one and double-six to appear. Craps enthusiasts would be wise to memorize this table:
In backgammon, another game in which the players throw two dice, the numbers on each die may be either added together or considered separately. This means, for example, that, when two dice are thrown, a 5 can appear in fifteen different ways:
The probability of a five-throw is 15/36, or about 42% I5
Semantics are important here. As Cardano put it, the probability of an outcome is the ratio of favorable outcomes to the total opportunity set. The odds on an outcome are the ratio of favorable outcomes to unfavorable outcomes. The odds obviously depend on the probability, but the odds are what matter when you are placing a bet.
If the probability of a five-throw in backgammon is 15 five-throws out of every 36 throws, the odds on a five-throw are 15 to 21. If the probability of throwing a 7 in craps is one out of six throws, the odds on throwing a number other than 7 are 5 to 1. This means that you should bet no more than $1 that 7 will come up on the next throw when the other fellow bets $5 that it won't. The probability of heads coming up on a coin toss are 50/50, or one out of two; since the odds on heads are even, never bet more than your opponent on that game. If the odds on a long-shot at the track are 20-to-1, the theoretical probability of that nag's winning is one out of 21, or 4.8%, not 5%.
In reality, the odds are substantially less than 5%, because, unlike craps, horse racing cannot take place in somebody's living room. Horse races require a track, and the owners of the track and the state that licenses the track all have a priority claim on the betting pool. If you restate the odds on each horse in a race in terms of probabilities-as the 20-to-1 shot has a p
robability of winning of 4.8%-and add up the probabilities, you will find that the total exceeds 100%. The difference between that total and 100% is a measure of the amount that the owners and the state are skimming off the top.
We will never know whether Cardano wrote Liber de Ludo Aleae as a primer on risk management for gamblers or as a theoretical work on the laws of probability. In view of the importance of gambling in his life, the rules of the game must have been a primary inspiration for his work. But we cannot leave it at that. Gambling is an ideal laboratory in which to perform experiments on the quantification of risk. Cardano's intense intellectual curiosity and the complex mathematical principles that he had the temerity to tackle in Ars Magna suggest that he must have been in search of more than ways to win at the gaming tables.
Cardano begins Liber de Ludo Aleae in an experimental mode but ends with the theoretical concept of combinations. Above its original insights into the role of probability in games of chance, and beyond the mathematical power that Cardano brought to bear on the problems he wanted to solve, Liber de Ludo Aleae is the first known effort to put measurement at the service of risk. It was through this process, which Cardano carried out with such success, that risk management evolved. Whatever his motivation, the book is a monumental achievement of originality and mathematical daring.
But the real hero of the story, then, is not Cardano but the times in which he lived. The opportunity to discover what he discovered had existed for thousands of years. And the Hindu-Arabic numbering system had arrived in Europe at least three hundred years before Cardano wrote Liber de Ludo Aleae. The missing ingredients were the freedom of thought, the passion for experimentation, and the desire to control the future that were unleashed during the Renaissance.
The last Italian of any importance to wrestle with the matter of probability was Galileo, who was born in 1564, the same year as William Shakespeare. By that time Cardano was already an old man.16 Like so many of his contemporaries, Galileo liked to experiment and kept an eye on everything that went on around him. He even used his own pulse rate as an aid in measuring time.
One day in 1583, while attending a service in the cathedral in Pisa, Galileo noticed a lamp swaying from the ceiling above him. As the breezes blew through the drafty cathedral, the lamp would swing irregularly, alternating between wide arcs and narrow ones. As he watched, he noted that each swing took precisely the same amount of time, no matter how wide or narrow the arc. The result of this casual observation was the introduction of the pendulum into the manufacture of clocks. Within thirty years, the average timing error was cut from fifteen minutes a day to less than ten seconds. Thus was time married to technology. And that was how Galileo liked to spend his time.
Nearly forty years later, while Galileo was employed as the First and Extraordinary Mathematician of the University of Pisa and Mathematician to His Serenest Highness, Cosimo II, the Grand Duke of Tuscany, he wrote a short essay on gambling "in order to oblige him who has ordered me to produce what occurs to me about the prob lem."17 The title of the essay was Sopra le Scoperte dei Dadi (On Playing Dice). The use of Italian instead of Latin suggests that Galileo had no great relish for a topic that he considered unworthy of serious consideration. He appears to have been performing a disagreeable chore in order to improve the gambling scores of his employer, the Grand Duke.
In the course of the essay, Galileo retraces a good deal of Cardano's work, though Cardano's treatise on gambling would not be published for another forty years. Yet Galileo may well have been aware of Cardano's achievement. Florence Nightingale David, historian and statistician, has suggested that Cardano had thought about these ideas for so long that he must surely have discussed them with friends. Moreover he was a popular lecturer. So mathematicians might very well have been familiar with the contents of Liber de Ludo Aleae, even though they had never read it.18
Like Cardano, Galileo deals with trials of throwing one or more dice, drawing general conclusions about the frequency of various combinations and types of outcome. Along the way, he suggests that the methodology was something that any mathematician could emulate. Apparently the aleatory concept of probability was so well established by 1623 that Galileo felt there was little more to be discovered.
Yet a great deal remained to be discovered. Ideas about probability and risk were emerging at a rapid pace as interest in the subject spread through France and on to Switzerland, Germany, and England.
France in particular was the scene of a veritable explosion of mathematical innovation during the seventeenth and eighteenth centuries that went far beyond Cardano's empirical dice-tossing experiments. Advances in calculus and algebra led to increasingly abstract concepts that provided the foundation for many practical applications of probability, from insurance and investment to such far-distant subjects as medicine, heredity, the behavior of molecules, the conduct of war, and weather forecasting.
The first step was to devise measurement techniques that could be used to determine what degree of order might be hidden in the uncertain future. Tentative efforts to devise such techniques were under way early in the seventeenth century. In 1619, for example, a Puritan minister named Thomas Gataker published an influential work, Of the Nature and Use of Lots, in which he argued that natural law, not divine law, determined the outcome of games of chance.19 By the end of the seventeenth century, about a hundred years after the death of Cardano and less than fifty years after the death of Galileo, the major problems in probability analysis had been resolved. The next step was to tackle the question of how human beings recognize and respond to the probabilities they confront. This, ultimately, is what risk management and decision-making are all about and where the balance between measurement and gut becomes the focal point of the whole story.
either Cardano nor Galileo realized that he was on the verge of articulating the most powerful tool of risk management ever to be invented: the laws of probability. Cardano had proceeded from a series of experiments to some important generalizations, but he was interested only in developing a theory of gambling, not a theory of probability. Galileo was not even interested in developing a theory of gambling.
Galileo died in 1642. Twelve years later, three Frenchmen took a great leap forward into probability analysis, an event that is the subject of this chapter. And less than ten years after that, what had been just a rudimentary idea became a fully developed theory that opened the way to significant practical applications. A Dutchman named Huygens published a widely read textbook about probability in 1657 (carefully read and noted by Newton in 1664); at about the same time, Leibniz was thinking about the possibility of applying probability to legal problems; and in 1662 the members of a Paris monastery named Port-Royal produced a pioneering work in philosophy and probability to which they gave the title of Logic. In 1660, an Englishman named John Graunt published the results of his effort to generalize demographic data from a statistical sample of mortality records kept by local churches. By the late 1660s, Dutch towns that had traditionally financed themselves by selling annuities were able to put these policies on a sound actuarial footing. By 1700, as we mentioned earlier, the English government was financing its budget deficits through the sale of life annuities.
The story of the three Frenchmen begins with an unlikely trio who saw beyond the gaming tables and fashioned the systematic and theoretical foundations for measuring probability. The first, Blaise Pascal, was a brilliant young dissolute who subsequently became a religious zealot and ended up rejecting the use of reason. The second, Pierre de Fermat, was a successful lawyer for whom mathematics was a sideline. The third member of the group was a nobleman, the Chevalier de Mere, who combined his taste for mathematics with an irresistible urge to play games of chance; his fame rests simply on his having posed the question that set the other two on the road to discovery.
Neither the young dissolute nor the lawyer had any need to experiment in order to confirm their hypotheses. Unlike Cardano, they worked inductively in creating for the
first time a theory of probability. The theory provided a measure of probability in terms of hard numbers, a climactic break from making decisions on the basis of degrees of belief.
Pascal, who became a celebrated mathematician and occasional philosopher, was born in 1623, just about the time Galileo was putting the finishing touches on Sopra le Scoperte dei Dadi. Born in the wake of the religious wars of the sixteenth century, Pascal spent half his life torn between pursuing a career in mathematics and yielding to religious convictions that were essentially anti-intellectual. Although he was a brilliant mathematician and proud of his accomplishments as a "geomaster," his religious passion ultimately came to dominate his life.'
Pascal began life as a child prodigy. He was fascinated with shapes and figures and discovered most of Euclidean geometry on his own by drawing diagrams on the tiles of his playroom floor. At the age of 16, he wrote a paper on the mathematics of the cone; the paper was so advanced that even the great Descartes was impressed with it.
This enthusiasm for mathematics was a convenient asset for Pascal's father, who was a mathematician in his own right and earned a comfortable living as a tax collector, a functionary known at the time as a tax farmer. The tax farmer would advance money to the monarch-the equivalent of planting his seeds-and then go about collecting it from the citizenry-the equivalent of gathering in a harvest whose ultimate value, as with all farmers, he hoped would exceed the cost of the seeds.
While Pascal was still in his early teens, he invented and patented a calculating machine to ease the dreary task of adding up M. Pascal's daily accounts. This contraption, with gears and wheels that went forward and backward to add and subtract, was similar to the mechanical calculating machines that served as precursors to today's electronic calculators. The young Pascal managed to multiply and divide on his machine as well and even started work on a method to extract square roots. Unfortunately for the clerks and bookkeepers of the next 250 years, he was unable to market his invention commercially because of prohibitively high production costs.