The theory of probability is often said to have begun with the work of two Frenchmen, Blaise Pascal (1623–62) and Pierre de Fermat (1601–65). They were both extremely successful mathe- maticians. Each of them made many discoveries in a variety of mathematical disciplines, but neither Fermat nor Pascal was pri- marily a mathematician. Both were mathematical hobbyists; fortu- nately, they were brilliant hobbyists.
Pierre de Fermat was 22 years older than Pascal. He studied law at the University of Toulouse and later found work with the government in the city of Toulouse. This allowed him to work as a lawyer and to pursue the many interests that he had outside the law. When the law courts were in session he was busy with the practice of law. When the courts were out of session he studied mathematics, literature, and lan- guages. Fermat knew many languages, among them Greek, Latin, Spanish, Italian, and, of course, French. He was well liked. By all accounts Fermat was polite and considerate and well educated, but beneath his genteel exterior he was passionately curious.
Mathematics is a difficult subject to pursue in isolation. The ideas involved can be conceptually difficult, and the solutions can be technically difficult. It is easy to get bogged down with details and miss the forest for the trees. To keep one’s mind fresh it helps to have access to other people with similar interests. For Fermat, “keeping fresh” meant sending letters to accomplished mathemati- cians. He maintained a lively correspondence with many of the best mathematicians of his time. The letters, many of which were pre- served, show a modest and inquisitive man in a serious and sus- tained search for mathematical truth.
In contrast to Fermat, Blaise Pascal spent his teenage years gleaning his mathematical education from face-to-face contact with some of the finest mathematicians in Europe. He accomplished this by attending one of the most famous math “clubs” in the history of the subject.
In France and Italy during the time of Fermat and Pascal, and even during the time of Cardano, there existed many formal and informal groups of like-minded individuals who met together to discuss new ideas in science and mathe-
matics. Meetings were held more or less regularly. One of the most famous of these groups met each week in Paris, Pascal’s hometown, at the house of Marin Marsenne. Marsenne was a priest with a love of science, mathematics, and music. He was a prolific writer and corresponded with many of the leading mathe- maticians and scientists of his day, but it was the meetings, held weekly at his house, that made him well known throughout Europe. Some of the finest mathematicians and scientists of the time spent one evening each week at what came to be known as the Marsenne Academy. They talked, they argued, and they learned. Pierre de Fermat, who lived in far-away Toulouse, was not a mem- ber, but another mathematician, Etienne Pascal, was frequently in attendance. In addition to his attendance at the academy, he and Fermat corresponded on a number of subjects. Although Etienne Pascal was a good mathematician, he is best remembered today as the father of Blaise Pascal.
Etienne Pascal, as did Fermat, worked as a civil servant, but his principal interest was his son’s education. Initially, he instructed Blaise in languages and literature. He would not teach him
Ancient dice and a shaker made of bone (Museum of London/Topham- HIP/The Image Works)
mathematics, because he did not want to overwork his son. It was not until the younger Pascal began to study geome- try on his own that his father relented and began to teach him math as well. Blaise Pascal was 12 when he began to receive instruction in mathematics. By the time he was 14 years of age he was accompanying his father to the get-togethers at Father Marsenne’s house.
The meetings had a pro- found effect on Blaise Pascal’s thinking. By the time he was 16 he had made an important discovery in the new field of projective geometry. (The mathematician who founded the field of projective geometry, Gérard (or Girard) Desargues, attended the meetings regularly, and Pascal’s discovery was an extension of the work of Desargues.) The younger Pascal’s interests changed quickly, however, and he soon stopped studying geometry. By the time he was 18 he was drawing attention to himself as the inventor of a mechanical calculator, which he created to help his father perform calculations in his capacity as a government official. The Pascaline, as it came to be called, was neither reliable nor cheap, but he made several copies and sold some of them. These calculators made a great impression on Pascal’s contemporaries, and several later calcula- tors incorporated a number of Pascal’s ideas into their design.
As an adult Pascal was acquainted with a French nobleman, the chevalier de Méré, a man who loved to gamble. Pascal and de Méré discussed the mathematical basis for certain problems asso- ciated with gambling. Pascal eventually turned to Fermat for help Blaise Pascal. His brief exchange of
letters with Pierre de Fermat opened up a new way of thinking about random processes. (Salaber/The Image Works)
in the solution of these problems. In 1654, Fermat and Pascal began a famous series of letters about games of chance.
Some of the problems that Pascal and Fermat discussed con- cerned “the division of stakes” problem. The idea is simple enough. Suppose that two players place equal bets on a game of chance. Suppose that one player pulls ahead of the other and then they decide to stop the game before it has reached its conclusion. How should they divide the stakes? If one player is ahead then it is unreasonable to divide the stakes in half since the player who is ahead would “probably” have won. As every gambler knows, how- ever, being ahead in a game of chance is no guarantee of a win: In fact, sometimes the player who is behind eventually wins anyway. Nevertheless, over the long run the player who is ahead wins more often than the player who is behind. The division of the stakes should reflect this. This problem involves several important prob- ability concepts and may have been inspired by ideas outside the field of gambling. (See the sidebar.)
In their letters Pascal and Fermat solve multiple versions of this type of gambling problem. They began with problems that involve two players and a single die. Later, they considered three-player games, but they did not limit themselves to the division of stakes problem. They also answered questions about the odds of rolling a particular number at least once in a given number of rolls. (What, for example, are the odds of rolling a 6 at least once in eight rolls of a die? See the sidebar Cardano’s Mistake earlier in this chapter for the solution to a closely related problem.) Their letters reflect a real excitement about what they were doing.
Unfortunately, Pascal and Fermat corresponded for only several months about games of chance, and then Pascal stopped working in mathematics altogether. He joined a religious order and gave up mathematics for the rest of his life. Several years later, Fermat sent Pascal one final letter offering to meet him halfway between their homes to visit, but Pascal refused. In a few more years both men were dead.
The sophistication of Fermat and Pascal’s work far surpassed that of the work of Cardano and Galileo. Previously, Cardano had asserted that what he had discovered about a single die was
interesting from a theoretical viewpoint but was worthless from a practical point of view. It is true that neither his discoveries nor any subsequent discoveries enable a gambler to predict which