Mathematics and the Real World: The Remarkable Role of Evolution in the Making of Mathematics (2014)

CHAPTER V. THE MATHEMATICS OF RANDOMNESS

36. PROBABILITY AND GAMBLING IN ANCIENT TIMES

The clearest expression of randomness in human behavior over thousands of years of human history is connected to gambling and games of chance. There is a wealth of evidence about games of chance in ancient times. The small anklebones of sheep, called astragals, were used by the Egyptians and the Assyrians for games of chance. The bones were thrown randomly, and bets were made on which side of the bone would appear on top (the bone had four sides), comparable to throwing dice in our days. The bone could fall on any of the four sides, but the chances of its falling on each side were not the same and changed from one bone to another. Archaeological finds from about six thousand years ago show evidence of these games of chance in ancient civilizations, including bones that had been polished and shaped for purposes of the game. The ancient Greeks adopted the game, and the shape of the bone appears in statues of men and women playing a game of rolling the bone. The Romans also played a game with such bones, which were called tali.

The point of interest from a mathematical aspect is that the ancients knew how to draw conclusions regarding what we would today call the chances of the bone falling on each of the sides. The evidence is indirect and is derived from tables of numbers that correspond with prizes for the correct guess as to the uppermost side. These calculations were made with no awareness of the concept of chance or probability of any particular result. These concepts appeared only in the seventeenth century. We do not know how the ancients calculated the numbers. It is reasonable to assume that they were calculated by observation and using intuition acquired by watching very many instances of throws of the bone, but there is no evidence regarding an ordered method used to calculate the prizes.

Later, at the time of the Greeks and more commonly among the Romans, the six-sided dice that we know today appeared, as well as dice with other geometrical shapes, such as four-sided pyramids, each side of which was an equilateral triangle. Various materials were used for these dice, including animal bones, stones, ivory, and lead. Much effort was invested in polishing and working the dice to obtain maximum symmetry, interpreted of course as meaning that there were equal chances for a die to fall on any of its sides. Then the faces of the six-sided dice were given the numbers from one to six, and the object of the most common game, still common today, was to obtain the highest score in rolling the die. These games of chance captivated both the ordinary populace as well as the rulers in Greece and Rome. The games are mentioned in the mythology as well as in reports of rulers who were addicted and who would hire aids to carry the accessories for games of dice wherever they went and to calculate the ruler's winnings or losses. Gambling and the use of stones for throwing to create randomness were so common that Judaism found it necessary to prohibit those activities explicitly, as is stated in Deuteronomy 18:10–11: “Let no one be found among you who sacrifices his son or daughter in the fire, who practices divination or sorcery, interprets omens, engages in witchcraft, or casts lots, or who is a medium or spiritualist or who consults the dead,” where “casting lots” means throwing cubes or dice and guessing the outcome. The commandment not to gamble shows that gambling was a social problem even then!

In contrast, there are instances in the Bible of the use of randomness as a positive mechanism. For example, Proverbs 18:18 says, “The lot causes contentions to cease, and parts between the mighty.” This means that the casting of lots can resolve disputes between litigants. The Torah also recounts several stories in which randomness served as an instrument with which to reach fair decisions. When the Land of Israel was divided among the tribes it says (Numbers 33:54): “And you shall divide the land by lot for an inheritance among your families: and to the more you shall give the more inheritance, and to the fewer you shall give the less inheritance: every man's inheritance shall be in the place where his lot falls; according to the tribes of your fathers you shall inherit.” “The lot” means casting lots, that is, the allocation of the land as inheritance between the tribes was carried out by lots. The Torah does not specify the method used to cast the lots, but the Talmud (the oral law eventually written down, the Mishna and the Gemara, interpreting and elaborating on the Bible) describes at length how the lots were cast. Note that the Mishna and the Gemara were written almost two thousand years ago. The procedure of casting the lots indicates (see, e.g., Bava Batra 122, Yerushalmi, Yoma 4.1) that two pitchers were shown to the people. In one were the names of the tribes, and in the other the names of the parcels of land. One name was taken at random from each pitcher, and that combination determined the allocation of the land. One might ask why wasn't one pitcher enough, holding the names of the tribes, and then for each parcel of land one name could have been drawn. From the mathematical aspect of the law of chance of today, there is no difference between the two methods. That was apparently understood even at that time, and the interpretations of the Talmud explain that the intention was to reinforce the fairness of the method (or in less politically correct parlance, in that way it was harder to cheat). In any event we see that the Bible and the commentaries saw drawing lots as a fair system.

This understanding also appeared in other cultures, such as in Greece. The Agora museum in Athens has an exhibit of a carved stone with a network of holes. The stone was used for selecting jurors for court cases held in the city. In the first stage, the men of the city would each insert a wood chip into a hole. Then the representative of the city, who had not been present at the first stage of the procedure, would come and randomly break off a number of the chips corresponding to the required number of jurors. Whoever's chip was broken had to serve as a juror for that day. In the next chapter, on the mathematics of human behavior, we will expand further on the use of a random process as a mechanism for achieving fairness.

In the two examples above, and in many other references to randomness both to achieve fairness as well as in relation to games of chance, the reference was based on intuitive understanding with no logical mathematical foundation, despite the fact that the logical approach and the use of axioms for mathematical analysis were quite developed. For some reason the contemporary scientists did not consider probability to be worthy of mathematical analysis. The notion of “probable” was used as early as in the days of Aristotle, but no proper attempt was made to develop or even to formulate the relevant mathematics. The absence of mathematical analysis of probability persisted until the beginning of the modern era. A number of factors resulted in the growing interest in the nature of the effects of probability, interest that eventually brought about the beginning of mathematical probability theory.

The popularity of games of chance did not wane for many years, and gambling houses spread all over Europe in the fifteenth and sixteenth centuries. The gamblers included some well-known mathematicians who apparently wanted to exploit their arithmetic abilities to become wealthy. The concept of chance or probability did not exist yet, but preliminary questions about the idea of probability had been asked; for example, out of a given number of throws, how often would two dice show the desired pair of sixes? Galileo was asked why, in a game consisting of throwing three dice, gamblers prefer to bet that the sum of the upper faces would be eleven rather than twelve, when both eleven and twelve can result from the same number of combinations of smaller numbers. In the same way they would prefer to bet on a total of ten rather than nine. Galileo's answer, which was correct, was that the number of ways in which the desired total can be described as a sum of three numbers between one and six is not important; what matters is the relative frequency of times the given total would appear when the three dice are thrown. The two computations are not the same. That was a mathematical explanation of the punters’ behavior, behavior that developed through experience.

The Italian mathematician Gerolamo Cardano (1501–1576), of Pavia, near Milan, was foremost in the development of formulae dealing with questions like the above. Cardano studied at the University of Padua, and he was a physician, an astrologer, a mathematician, and also an inveterate gambler. In the field of mathematics he was well known in particular for his development of methods of solving cubic and quartic equations. He fell out with Niccolò Tartaglia (1499–1557), among others, who showed Cardano his method of solving those equations. Cardano did not hide the fact that he had learned the method from Tartaglia, but the latter claimed that he had made Cardano swear not to publicize his method, as doing so would harm him. The ability to solve equations was a way of making money by winning public equation-solving competitions.

Cardano also suffered from shortage of money, and he tried to get it via games of chance and gambling, and for that purpose he developed mathematical methods of calculating what is today referred to as the relative frequency of numbers coming up in throws of dice, for example, as we mentioned above, calculating the number of times a pair of sixes would be thrown or how often the sum of the two upper faces would exceed ten. Cardano gathered these methods and other studies related to gambling into notes for a book, but it was not published until after his death. Cardano's and his contemporaries’ methods were limited to methods of calculation; in other words, they translated intuition into arithmetic without any logical mathematical basis. This changed very soon.