Rene Ritson
"The loser, when the game of dice breaks up,
lingers despondent, and repeats the throws
to learn in grief, what made his fortunes droop"
Dante’s Divine Comedy
In 1991, the application of probability to observations of the volcanic activity of Mount Pinatubo in the Phillippines led to such accurate forecasting that fewer than 500 people died in the biggest volcanic eruption for 80 years in that area. In 1994, the National Lottery was introduced in Britain offering a 1 in 13,983,816 chance of winning millions of pounds. Two different situations, but these are two sides of the same coin.
Probability is one of the youngest branches of mathematics, being only marginally younger than calculus, but the development of probability from early beginnings was very much slower than that of calculus. In comparison with calculus, the beginnings of probability can be traced back to about 5000 BC, or even earlier, and there is ample archaeological evidence that games of chance were played in ancient times in many different parts of the world.
An astragalus, a small bone in the hock of a hooved animal, is shaped in such a way that it has only four faces on which it can rest when thrown and is almost symmetrical. Archaeological excavations on various sites have produced a preponderance of unmodified astragali among other animal bones which had obviously been adapted as tools, indicating that they had been used in games of chance, but there seems to have been no standard method of marking or enumerating the faces of the astragali. As with a modern die, the score from a throw was that on the uppermost face. It is believed that one of the games of ancient Greece was to throw four astragali and note the scores on the uppermost faces with the aim of trying to obtain "the throw of Venus" in which all four faces were different. Other evidence of games of chance has come from finds which included throwing sticks which were concave on one long side and convex on the other, simple roulettes, and cubical dice, though these appear to have developed much later than the use of astragali. What is surprising is that dice were designed in so many different parts of the world so early in history when there could have been no communication between communities so far apart but, however this happened, dice were here to stay.
Dice and board games have been found in excavations in tombs in different parts of Egypt, and a multitude of Roman dice have been found from different parts of the Roman empire. It would appear that the Romans were inveterate gamblers to such an extent that laws to restrict gambling were introduced, but they seem to have had little effect, a pattern which was repeated in different parts of the world for centuries to come. Ironically, it was the interest of gamblers which eventually led to a quantification of chance and hence to the mathematics of probability.
By the 15th century gambling with dice was a matter of using either two or three dice and it was then that attempts were made to enumerate the possible scores from such combinations of dice. One such attempt for three dice was by Benvenuto d’Imola, who wrote: "With three dice, three is the smallest number which can be thrown, and that only when three aces turn up; four can only happen in one way, namely as two and two aces . . .", from which it is clear that he did not consider that two and two aces could be 1+1+2, 1+2+1 or 2+1+1, nor was this the last time historically that such a mistake was to occur. About 50 years later, an Italian scholar, Geralamo Cardano, who was primarily a physician but was interested in mathematics, science and astrology, wrote a book on games of chance which included a chapter in which he correctly enumerated the throws of two dice and of three dice and showed clearly how each could occur. This book was not published until long after his death and could be considered as one of many missed opportunities in the development of mathematical probability.
During Cardano’s lifetime, another Italian, Luca Pacioli, produced a treatise on the mathematical knowledge of the time which included a section on unusual problems, one of which was:
A team plays ball so that a total of 60 points is required to win and the stakes are 22 ducats. Due to circumstances, they cannot finish the game and one side has 50 points and the other 30. What share of the prize money belongs to each side?
Pacioli’s solution was simply to divide the prize money in the same proportion as the points already scored, but Cardano pointed out that this did not take account of the possible outcomes of the games yet to be played. A third mathematician, Tartaglia, joined the fray and commented: "It is a subject making little sense and will be substantial cause for litigation", but his own solution was no better. He reasoned that the difference between the two scores was one-third of the score needed to win so the side with 50 points should take two-thirds of the stake and the other side one-third.
This problem subsequently became known as "the problem of points" when it was raised again by the gambler Chevalier de Mé ré in the 17th century, who put the problem to Pierre de Fermat and sparked a correspondence between Fermat and Blaise Pascal. It took very little time for each to solve the problem, though by different methods. Fermat’s preference was to look at the ways in which play might continue, while Pascal calculated the expected shares after each individual game. When these two went on to consider a problem of points with three players it was Fermat who produced a correct solution.
This period was a time of considerable mathematical activity, with the development of algebra changing ways of thinking, but, above all, providing a tool which enabled problems to be generalised. The availability of this tool, combined with the fact that Fermat communicated freely with other mathematicians of his day, helped to ensure that probability was to become accepted as a respectable branch of mathematics. The astronomer Christiaan Huygens heard of the work of Pascal and Fermat on the problem of points but, knowing nothing of the details of their solutions, he worked on the problem himself and produced a solution along the same lines as Pascal. A few years later, Huygens produced a treatise on calculations of chance which probably did much to popularize interest in mathematical probability and many famous mathematicians became involved in probabilistic discussions, though not without a sprinkling of misconceptions. Roberval, better known for his work in the field of geometry, and D’Alembert, a century later, both argued that the probability of getting at least one head in two tosses of an unbiased coin was 2 in 3 since the only possibilities were two heads, two tails or one head and one tail. D’Alembert also believed that after a long run of heads the chance of tail became more likely.
This general interest in probability was soon to be overshadowed by the development of calculus which, once mastered, was seen as a powerful tool capable of advancing almost all areas of mathematics. While the mathematicans of the day were learning how to use calculus there were changes in social conditions which were to have subsequent implications for the applications of probability. As early as 1592, registers of the annual number of deaths in London began to be kept, though somewhat irregularly until 1603, when the plague was sweeping through the country. Initially for the purpose of monitoring the progress of the plague, Bills of Mortality were published every week on Thursday and the record for the whole year was published on the last Thursday before Christmas day on a regular basis from 1603. The publication of annual records continued until well into the 19th century. John Graunt was a haberdasher by trade but he used the annual records to determine the population growth in the capital and in 1662 published a treatise, "Natural and Political Observations made upon the Bills of Mortality" , based on the data available. This was the beginning of a statistical process which was to contribute to the development of a different aspect of probability.
After the publication of Graunt’s treatise using the data recorded in London, other European cities began to make use of their own population data and, in 1693, the astronomer, mathematician and physicist Edmund Halley produced the first life insurance tables based on his study of Graunt’s treatise and Bills of Mortality for Dublin and for Breslau. In producing these tables, Halley showed how to use known data to calculate the chance of an individual surviving a given number of years at any particular age. During approximately the same period of time as the works by Graunt and Halley, there were attempts in the Netherlands to calculate reasonable prices for life annuities, a task which depended on probabilities associated with life expectancy.
It was inevitable that these early attempts to apply probability to the calculation of life expectancy would come under close scrutiny by the mathematical fraternity and, not least among these scrutineers, were Gottfried Leibniz and Jacques Bernoulli, one of three brothers in the first of three generations of the Bernoulli mathematical dynasty. Leibniz and Bernoulli were in frequent correspondence and discussion, initially through Bernoulli’s desire to master Leibnizian calculus, but their discussions were mathematically wide-ranging and included the probabilistic assumptions made by Graunt, Halley and others in their attempts to calculate life expectancy.
The overall effect of this social influence was to return probability nearer to the forefront of mathematical interest and attract the attention of many who, like Abraham de Moivre, are probably better known for their other work. By the early 18th century, the theory of probability was mathematically respectable and De Moivre produced his "Doctrine of Chance" in 1718, with a later edition in 1738 and a posthumous edition published in 1756. The first part was devoted to solving problems of chance which had been put to him by gambler friends and the final part was devoted to the calculations of annuities on lives. However, what is regarded as the first substantial publication on the theory of probability was the work of Jacques Bernoulli, who also introduced the notion of taking certainty to be a unit so that the measure of probability is then a number between 0 and 1.
During the next hundred years, probability became more and more academic and remained in the domain of research, with the most comprehensive work on the subject being a very academic treatise by Pierre Simon de Laplace, but this was only comprehensible to scholars who already had some knowledge of the calculus of probability. The need for a textbook-type treatment aimed at the ordinary scholar was recognised by Augustus de Morgan in a commentary on the theory of probability which was published in "The Dublin Review" in 1837. The alternative to a textbook was the presentation of the necessary foundations from which study of the more academic works could proceed and it was this route that he chose. De Morgan was the first professor of mathematics at University College, London and from a review of mathematics courses in academic institutions in London during the reign of Queen Victoria, the content and presentation of De Morgan’s lectures give a clear impression of a teacher ahead of his time. His emphasis on understanding mathematical principles, his use of problem solving and his dislike of having students compete against each other in examinations as a source of stimulus made his teaching stand out above the style of the times. In response to those who maintained that there could be no stimulus without competition he felt that the student’s own pleasure of learning should be sufficient stimulus adding that " ... if a teacher cannot make them feel this, he does not deserve the name of teacher". Although not included in the college calendars there is evidence from students’ memoirs that he also taught some probability theory to the more able mathematics students.
With De Morgan’s bridge in place, probability calculus gradually ceased to be confined to the field of research and was widened to allow alternative techniques which would satisfy the ever-increasing demands for mathematically rigorous methods as well as its ever widening applications. From then on, probability has gradually invaded the realms of education, industry and almost every aspect of daily life.