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9/11 Remembered
A Brief History of Probability
Posted by Bill Abrams, Coeditor SecondMoment

Adapted from An Introduction to Mathematical Statistics and its Applications by Richard J. Larsen and Morris L. Marx

No one knows where or when the notion of chance first arose. Nevertheless, evidence linking early humans with devices for generating random events is plentiful. Archaeological digs throughout the ancient world consistently turn up a curious overabundance of astragali, the heel bones of sheep and other vertebrates. Why should the frequencies of these bones be so disproportionately high? One could hypothesize that our forebears were fanatical foot fetishists, but two other explanations seem more plausible—the bones were used for religious ceremonies and for gambling.

Astragali have six sides but are not symmetrical. Those found in excavations typically have their sides numbered or engraved. For many ancient civilizations, astragali were the primary mechanism through which oracles solicited the opinions of their gods. In Asia Minor, for example, it was customary in divination rites to roll, or cast, five astragali. Each possible configuration was associated with the name of a god and carried with it the sought-after advice. An outcome of (1,3,3,4,4), for instance, was said to be the throw of the savior Zeus, and was taken as a sign of encouragement. A (4,4,4,6,6), on the other hand, the throw of the child-eating Cronos, would send everyone scurrying for cover.

Gradually, over thousands of years, astragali were replaced by dice, and the latter became the most common means of generating random events. Pottery dice have been found in Egyptian tombs built before 2000 B.C, and by the time Greek civilizations was in full flower, dice were everywhere. (Loaded dice have also been found from antiquity. While mastering the mathematics of probability would prove to be a formidable task for our ancestors, they quickly learned how to cheat.)

The historical record blurs the distinctions initially drawn between divination ceremonies and recreational gaming. Among more recent societies, though, gambling emerged as a distinct entity, and its popularity was irrefutable. The Greeks and Romans were consummate gamblers, as were the early Christians. Rules for many of the Greek and Roman games have been lost, but we can recognize the lineage of certain modern diversions in what was played during the Middle Ages. The most popular dice game of that period was called hazard, the name deriving from the Arabic al zhar, which means “a die.” Hazard is thought to have been brought to Europe by soldiers returning from the Crusades, and its rules are much like those of our modern-day craps. Cards were first introduced in the fourteenth century and immediately gave rise to a game known as Primero, and early form of poker. Board games such as backgammon were also popular during this period.

Given this rich tapestry of games and the obsession with gambling that characterized so much of the Western World, it may seem more than a little puzzling that a formal study of probability was not undertaken sooner than it was. The first instance of anyone conceptualizing probability in terms of a mathematical model occurred in the sixteenth century, which means that more than 2000 years of dice games, card games, and board games passed by before someone finally had the insight to write down even the simplest probabilistic abstractions.

Greek Philosophy and Early Christian Theology
Historians generally agree that, as a subject, probability got off to a rocky start because of its incompatibility with two of the most dominant forces in the evolution of our Western culture, Greek philosophy and early Christian theology. The Greeks were comfortable with the notion of chance, but it went against their nature to suppose that random events could be quantified in any useful fashion. They believed that any attempt to reconcile mathematically what did happen with what should have happened was, in their phraseology, an improper juxtaposition of the “earthly plane” with the “heavenly plane.”

Making matters worse was the antiempiricism that permeated Greek thinking. Knowledge, to them, was not something that should be derived by experimentation. It was better to reason out a question logically than to search for its explanation in a set of numerical observations. Together, these two attitudes had a deadening effect. The Greeks had no motivation to think about probability in any abstract sense, nor were they faced with the problems of interpreting data that might have pointed them in the direction of a probability calculus.

If the prospects for the study of probability were dim under the Greeks, they became even worse when Christianity broadened its sphere of influence. The Greeks and Romans at least accepted the existence of chance. They believed their gods to be either unable or unwilling to get involved in matters so mundane as the outcome of the roll of a die.

For the early Christians, though, there was no such thing as chance. Every event, no matter how trivial, was perceived to be a direct manifestation of God’s deliberate intervention. In the words of St. Augustine: “We say that those causes that are said to be by chance are not nonexistent but are hidden, and we attribute them to the will of the true God…” Taking Augustine’s position makes the study of probability moot, and it makes a probabilist a heretic. Not surprisingly, nothing of significance was accomplished in the subject for the next fifteen hundred years.

The Renaissance
It was in the sixteenth century that probability, like a mathematical Lazarus, arose from the dead. Orchestrating its resurrections was one of the most eccentric figures in the entire history of mathematics, Gerolamo Cardano. By his own admission, Cardano personified the best and worst of the Renaissance man. He was born in Pavia in 1501, but facts about his personal life are difficult to verify. He wrote an autobiography, but his penchant for lying raises doubts much of what he says.

Cardano was formally trained in medicine, but his interest in probability derived from his addiction to gambling. His love of dice and cards was so all-consuming that he is said to have once sold all his wife’s possessions just to get table stakes! Fortunately, something positive came out of Cardano’s obsession. He began looking for a mathematical model that would describe in an abstract way the outcome of a random event. What he eventually formalized is now called the classical definition of probability: If the total number of possible outcomes, all equally likely, associated with some actions is n and if m of those n result in the occurrence of some given event, then the probability of that event is m/n. Put another way, if a fair die is rolled, there are n = 6 possible outcomes. If the event “outcome is greater than or equal to 5” is the one in which we are interested, then m = 2 (the outcomes 5 and 6) and the probability of the even is 2/6, or 1/3.

Cardano had tapped into the most basic principle in probability. The model he discovered may seem trivial in retrospect, but it represented a giant step forward. His was the first recorded instance of anyone computing a theoretical, as opposed to an empirical, probability. Still, the actual impact of Cardano’s work was minimal. He wrote a book in 1525, but it was not published until 1663, and by then, the focus of the Renaissance, as well as interest in probability, had shifted from Italy to France.

More Gambling or The Problem of Points
The date cited by many historians as the “beginning” of probability is 1654. In Paris a well-to-do gambler, the Chevalier de Mere, asked several prominent mathematicians, including Blaise Pacal, a series of questions, the best known of which was the problem of points:

Two people, A and B, agree to play a series of fair games until one person has won six games. They each have wagered the same amount of money, the intention being that the winner will be awarded the entire pot. But suppose, for whatever reason, the series is prematurely terminated, at which point A has won five games and B three. How should the stakes be divided?

[The correct answer is that A should receive seven-eights of the total amount wagered. (Hint: suppose the contest was resumed, what scenarios would lead to A being the first person to win six games?)]

Pascal was intrigued by de Mere’s questions and shared his thoughts with Pierre Fermat, a Toulouse civil servant and probably the most brilliant mathematician in Europe. Fermat graciously replied, and from the now famous Pascal-Fermat correspondence came not only the solution to the problem of points but the foundation for more general results. More significantly, news of what Pascal and Fermat were working on spread quickly. Others got involved, of whom the best known was the Dutch scientists and mathematician Christiaan Huygens. The delays and the indifference that plagued Cardano a century earlier were not going to happen again.

Best remembered for his work in optics and astronomy, Huygens, early in his career, was intrigued by the problem of points. In 1657 he published De Ratiociniis in Aleae Ludo (Calculations in Games of Chance), a very significant work, far more comprehensive than anything Pascal and Fermat had done. For almost 50 years it was the standard “textbook” in the theory of probability. Huygens, of course, has supporters who feel that he should be credited as the founder of probability.

Almost all the mathematics of probability was still waiting to be discovered. What Huygens wrote was only the humblest of beginnings, a set of 14 Propositions bearing little resemblance to the topics we teach today. But the foundation was there. The mathematics of probability was finally on firm footing.