Seen in these terms, the Pascal-Fermat solution is clearly colored by the notion of risk management, even though they were not thinking explicitly in those terms. Only the foolhardy take risks when the rules are unclear, whether it be balla, buying IBM stock, building a factory, or submitting to an appendectomy.
But beyond the moral question, the solutions proposed by Pascal and Fermat lead to precise generalizations and rules for calculating probabilities, including cases involving more than two players, two teams, two genders, two dice, or coins with two sides. Their achievement enabled them to push the limits of theoretical analysis far beyond Cardano's demonstration that two dice of six sides each (or two throws of one die) would produce 62 combinations or that three dice would produce 63 combinations.
The last letter of the series is dated October 27, 1654. Less than a month later, Pascal underwent some kind of mystical experience. He sewed a description of the event into his coat so that he could wear it next to his heart, claiming "Renunciation, total and sweet." He abandoned mathematics and physics, swore off high living, dropped his old friends, sold all his possessions except for his religious books, and, a short while later, took up residence in the monastery of Port-Royal in Paris.
Yet traces of the old Blaise Pascal lingered on. He established the first commercial bus line in Paris, with all the profits going to the monastery of Port-Royal.
In July 1660, Pascal took a trip to Clermont-Ferrand, not far from Fermat's residence in Toulouse. Fermat proposed a meeting "to embrace you and talk to you for a few days," suggesting a location halfway between the two cities; he claimed bad health as an excuse for not wanting to travel the full distance. Pascal wrote back in August:
I can scarcely remember that there is such a thing as Geometry [i.e., mathematics]. I recognize Geometry to be so useless that I can find little difference between a man who is a geometrician and a clever craftsman. Although I call it the best craft in the world it is, after all, nothing else but a craft .... It is quite possible I shall never think of it again. 17
Pascal put together his thoughts about life and religion while he was at Port-Royal and published them under the title Pensees.'8 In the course of his work on that book, he filled two pieces of paper on both sides with what Ian Hacking describes as "handwriting going in all directions ... full of erasures, corrections, and seeming afterthoughts." This fragment has come to be known as Pascal's Wager (le pari de Pascal), which asks, "God is, or he is not. Which way should we incline? Reason cannot answer."
Here, drawing on his work in analyzing the probable outcomes of the game of balla, Pascal frames the question in terms of a game of chance. He postulates a game that ends at an infinite distance in time. At that moment, a coin is tossed. Which way would you bet-heads (God is) or tails (God is not)?
Hacking asserts that Pascal's line of analysis to answer this question is the beginning of the theory of decision-making. "Decision theory," as Hacking describes it, "is the theory of deciding what to do when it is uncertain what will happen."19 Making that decision is the essential first step in any effort to manage risk.
Sometimes we make decisions on the basis of past experience, out of experiments we or others have conducted in the course of our lifetime. But we cannot conduct experiments that will prove either the existence or the absence of God. Our only alternative is to explore the future consequences of believing in God or rejecting God. Nor can we avert the issue, for by the mere act of living we are forced to play this game.
Pascal explained that belief in God is not a decision. You cannot awaken one morning and declare, "Today I think I will decide to believe in God." You believe or you do not believe. The decision, therefore, is whether to choose to act in a manner that will lead to believing in God, like living with pious people and following a life of "holy water and sacraments." The person who follows these precepts is wagering that God is. The person who cannot be bothered with that kind of thing is wagering that God is not.
The only way to choose between a bet that God exists and a bet that there is no God down that infinite distance of Pascal's coin-tossing game is to decide whether an outcome in which God exists is preferablemore valuable in some sense-than an outcome in which God does not exist, even though the probability may be only 50-50. This insight is what conducts Pascal down the path to a decision-a choice in which the value of the outcome and the likelihood that it may occur will differ because the consequences of the two outcomes are different.*
If God is not, whether you lead your life piously or sinfully is immaterial. But suppose that God is. Then if you bet against the existence of God by refusing to live a life of piety and sacraments you run the risk of eternal damnation; the winner of the bet that God exists has the possibility of salvation. As salvation is clearly preferable to eternal damnation, the correct decision is to act on the basis that God is. "Which way should we incline?" The answer was obvious to Pascal.
Pascal produced an interesting by-product when he decided to turn over the profits from his bus line to help support the Port-Royal monastery.20 In 1662, a group of his associates at the monastery published a work of great importance, La logique, ou fart de penser (Logic, or the Art of Thinking), a book that ran to five editions between 1662 and 1668.t Although its authorship was not revealed, the primary-but not the sole-author is believed to have been Antoine Arnauld, a man characterized by Hacking as "perhaps the most brilliant theologian of his time."21 The book was immediately translated into other languages throughout Europe and was still in use as a textbook in the nineteenth century.
The last part of the book contains four chapters on probability that cover the process of developing a hypothesis from a limited set of facts; today, this process is called statistical inference. Among other matters, these chapters contain a "rule for the proper use of reason in determining when to accept human authority," rules for interpreting miracles, a basis of interpreting historical events, and the application of numerical measures to probability.22
The final chapter describes a game in which each of ten players risks one coin in the hope of winning the nine coins of his fellow players. The author then points out that there are "nine degrees of probability of losing a coin for only one of gaining nine."23 Though the observation is innocuous, the sentence has earned immortality. According to Hacking, this is the first occasion in print "where probability, so called, is measured.""
The passage deserves immortality for more reasons than that. The author admits that the games he has described are trivial in character, but he draws an analogy to natural events. For example, the probability of being struck by lightning is tiny but "many people ... are excessively terrified when they hear thunder. "25 Then he makes a critically important statement: "Fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event."26 Here is another major innovation: the idea that both gravity and probability should influence a decision. We could turn this assertion around and state that a decision should involve the strength of our desire for a particular outcome as well as the degree of our belief about the probability of that outcome.
The strength of our desire for something, which came to be known as utility, would soon become more than just the handmaiden of probability. Utility was about to take its place at the center of all theories of decision-making and risk-taking. It will reappear repeatedly in the chapters ahead.
Historians are fond of referring to near-misses-occasions when something of enormous importance almost happened but, for one reason or another, failed to happen. The story of Pascal's Triangle is a striking example of a near-miss. We have seen how to predict the probable number of boys or girls in a multi-child family. We have gone beyond that to predict the probable outcome of a World Series (for evenly matched teams) after part of the Series has been played.
In short, we have been forecasting! Pascal and Fermat held the key to a systematic method for calculating the probabilities of future events. Even though they did not tu
rn it all the way, they inserted the key into the lock. The significance of their pioneering work for business management, for risk management, and, in particular, for insurance was to be seized upon by others-for whom the Port-Royal Logic was an important first step. The idea of forecasting economic trends or of using probability to forecast economic losses was too remote for Pascal and Fermat to have recognized what they were missing. It is only with hindsight that we can see how close they came.
The inescapable uncertainty of the future will always prevent us from completely banishing the fates from our hopes and fears, but after 1654 mumbo jumbo would no longer be the forecasting method of choice.
e all have to make decisions on the basis of limited data. One sip, even a sniff, of wine determines whether the whole bottle is drinkable. Courtship with a future spouse is shorter than the lifetime that lies ahead. A few drops of blood may evidence patterns of DNA that will either convict or acquit an accused murderer. Public-opinion pollsters interview 2,000 people to ascertain the entire nation's state of mind. The Dow Jones Industrial Average consists of just thirty stocks, but we use it to measure changes in trillions of dollars of wealth owned by millions of families and thousands of major financial institutions. George Bush needed just a few bites of broccoli to decide that that stuff was not for him.
Most critical decisions would be impossible without sampling. By the time you have drunk a whole bottle of wine, it is a little late to announce that it is or is not drinkable. The doctor cannot draw all your blood before deciding what medicine to prescribe or before checking out your DNA. The president cannot take referendums of 100% of all the voters every month before deciding what the electorate wants-nor can he eat all the broccoli in the world before expressing his distaste for it.
Sampling is essential to risk-taking. We constantly use samples of the present and the past to guess about the future. "On the average" is a familiar phrase. But how reliable is the average to which we refer? How representative is the sample on which we base our judgment? What is "normal," anyway? Statisticians joke about the man with his feet in the oven and his head in the refrigerator: on the average he feels pretty good. The fable about the blind men and the elephant is famous precisely because each man had taken such a tiny sample of the entire animal.
Statistical sampling has had a long history, and twentieth-century techniques are far advanced over the primitive methods of earlier times. The most interesting early use of sampling was conducted by the King of England, or by his appointed proxies, in a ceremony known as the Trial of the Pyx and was well established by 1279 when Edward I proclaimed the procedure to be followed.'
The purpose of the trial was to assure that the coinage minted by the Royal Mint met the standards of gold or silver content as defined by the Mint's statement of standards. The strange word "pyx" derives from the Greek word for box and refers to the container that held the coins that were to be sampled. Those coins were selected, presumably at random, from the output of the Mint; at the trial, they would be compared to a plate of the King's gold that had been stored in a thricelocked treasury room called the Chapel of the Pyx in Westminster Abbey. The procedure permitted a specifically defined variance from the standard, as not every coin could be expected to match precisely the gold to which it was being compared.
A more ambitious and influential effort to use the statistical process of sampling was reported in 1662, eight years after the correspondence between Pascal and Fermat (and the year in which Pascal finally discovered for himself whether God is or God is not). The work in question was a small book published in London and titled Natural and Political Observations made upon the Bills Of Mortality. The book contained a compilation of births and deaths in London from 1604 to 1661, along with an extended commentary interpreting the data. In the annals of statistical and sociological research, this little book was a stunning breakthrough, a daring leap into the use of sampling methods and the calculation of probabilities-the raw material of every method of risk management, from insurance and the measurement of environmental risks to the design of the most complex derivatives.
The author, John Graunt, was neither a statistician nor a demographer-at that point there was no such thing as either.2 Nor was he a mathematician, an actuary, a scientist, a university don, or a politician. Graunt, then 42 years old, had spent his entire adult life as a merchant of "notions," such as buttons and needles.
Graunt must have been a keen businessman. He made enough money to be able to pursue interests less mundane than purveying merchandise that holds clothing together. According to John Aubrey, a contemporary biographer, Graunt was "a very ingenious and studious person ... [who] rose early in the morning to his Study before shoptime .... [V]ery facetious and fluent in his conversation."3 He became close friends with some of the most distinguished intellectuals of his age, including William Petty, who helped Graunt with some of the complexities of his work with the population statistics.
Petty was a remarkable man. Originally a physician, his career included service as Surveyor of Ireland and Professor of Anatomy and Music. He accumulated a substantial fortune as a profiteer during the wars in Ireland and was the author of a book called Political Arithmetick, which has earned him the title of founder of modern economics.4
Graunt's book went through at least five editions and attracted a following outside as well as inside England. Petty's review in the Parisian Journal des Scavans in 1666 inspired the French to undertake a similar survey in 1667. And Graunt's achievements attracted sufficient public notice for Charles II to propose him for membership in the newly formed Royal Society. The members of the Society were not exactly enthusiastic over the prospect of admitting a mere tradesman, but the King advised them that, "if they found any more such Tradesmen, they should be sure to admit them all, without any more ado." Graunt made the grade.
The Royal Society owes its origins to a man named John Wilkins (1617-1672), who had formed a select club of brilliant acquaintances that met in his rooms in Wadham College.5 The club was a clone of Abbe Mersenne's group in Paris. Wilkins subsequently transformed these informal meetings into the first, and the most distinguished, of the scientific academies that were launched toward the end of the seventeenth century; the French Academie des Sciences was founded shortly after, with the Royal Society as its model.
Wilkins later became Bishop of Chichester, but he is more interesting as an early author of science fiction embellished with references to probability. One of his works carried the entrancing title of The Discovery of a World in the Moone or a discourse tending to prove that 'tis probable there may be another habitable world in that planet, published in 1640. Anticipating Jules Verne, Wilkins also worked on designs for a submarine to be sent under the Arctic Ocean.
We do not know what inspired Graunt to undertake his compilation of births and deaths in London, but he admits to having found "much pleasure in deducing so many abstruse, and unexpected inferences out of these poor despised Bills of Mortality .... And there is pleasure in doing something new, though never so little."6 But he had a serious objective, too: "[T]o know how many people there be of each Sex, State, Age, Religious, Trade, Rank, or Degree, &c. by the knowing whereof Trade and Government may be made more certain, and Regular; for, if men know the People as aforesaid, they might know the consumption they would make, so as Trade might not be hoped for where it is impossible."7 He may very well have invented the concept of market research, and he surely gave the government its first estimate of the number of people available for military service.
Information about births and deaths had long been available in parish churches, and the City of London itself had started keeping weekly tallies from 1603 onward. Additional data were available in Holland, where the towns were financing themselves with life annuities-policies purchased for a lump sum that would pay an income for life to the owner of the policy, and occasionally to survivors. Churches in France also kept records of christenings and deaths.
Hacking reports t
hat Graunt and Petty had no knowledge of Pascal or Huygens, but, "Whether motivated by God, or by gaming, or by commerce, or by the law, the same kind of ideas emerged simultane ously in many minds."8 Clearly Graunt had chosen a propitious moment for publishing and analyzing important information about the population of England.
Graunt was hardly aware that he was the innovator of sampling theory. In fact, he worked with the complete set of the bills of mortality rather than with a sample. But he reasoned systematically about raw data in ways that no one had ever tried before. The manner in which he analyzed the data laid the foundation for the science of statistics.' The word "statistics" is derived from the analysis of quantitative facts about the state. Graunt and Petty may be considered the co-fathers of this important field of study.
Graunt did his work at a time when the primarily agricultural society of England was being transformed into an increasingly sophisticated society with possessions and business ventures across the seas. Hacking points out that so long as taxation was based on land and tillage nobody much cared about how many people there were. For example, William the Conqueror's survey known as the Domesday Book of 1085 included cadasters-registers of ownership and value of real property-but paid no heed to the number of human beings involved.
As more and more people came to live in towns and cities, however, headcounts began to matter. Petty mentions the importance of population statistics in estimating the number of men of military age and the potential for tax revenues. But for Graunt, who appears to have been a tradesman first, at a time of rising prosperity, political considerations were of less interest.
Against the Gods: The Remarkable Story of Risk Page 8