Mathematics and the Real World (30 page)

In experiments performed to show that animals react correctly to random situations, food was provided over a long period in one of two rooms, say
A
and
B
. The room was chosen randomly but with different probabilities. For example, room
A
would be chosen with a 40 percent probability, and room
B
with a 60 percent probability. A bell was sounded to indicate to the animals that food had arrived, without indicating in which room. If the animal went to the wrong room, it missed out on that meal.
Many subjects of the experiments, such as rats and pigeons, quickly discovered that the process was random and chose room
B
significantly more than room
A
.

One aspect of randomness is the adoption of strategies of random search for food, or random sampling of places to look for food. If the location of the food is random with known probabilities, a mathematical calculation can help to formulate optimal strategies of searching. Such a calculation shows that in many instances a random search is optimal. Extensive studies of food-search strategies among animals have shown that they do indeed adopt optimal random searches. That is not surprising. Even without learning and using the mathematics of optimal search, evolution nurtured those animals that adopted optimal-search policies. An animal that adopted the optimal-search strategy had an advantage in the evolutionary struggle, and natural selection gave preference to those animals that developed the ability to deal with randomness correctly.

Certain species of birds showed more sophisticated behavior related to randomness, such as, for example, the black-capped chickadee, which is the national bird of the states of Massachusetts and Maine in the United States. The chickadee, like other species, spends most of its time in thick shrubbery for safety, but it has to peck for its food in open spaces, and therefore from time to time it has to leave the shelter provided by the bushes. When searching for food it is thus exposed to the dangers of various predators, mainly larger birds of prey that cannot get into the bushes. If the chickadee leaves the bushes to search for food according to a set routine, the birds of prey would soon learn the pattern, and its chances of surviving would be low. Observations have revealed that it exits its shelter randomly, making it difficult for predators to predict when it will be exposed. The birds of prey also adopt a strategy of randomness, otherwise the chickadee would learn the hunters’ patterns and would leave its shelter only when there were none around. The chickadee's strategy of leaving the bushes, that is, the average frequency, the time it spends “outside,” and so on, takes into account the hunting strategy of the birds of prey. For example, if the average appearance of the bird of prey is short, the chickadee can allow itself more time away from its shelter. The ecologist Steven Lima, of Indiana State University,
carried out an interesting series of experiments (the results were published in 1985). Lima confronted the black-capped chickadee with a situation in which the parameters of the predator birds’ search strategy changed from time to time, for instance, the chance that it would be at a certain location. The chickadee quickly recognized the changes in the predators’ hunt and adjusted its own random parameters for leaving its shelter. In other words, in the evolutionary process, not only did a species of bird develop that knows how to behave in an environment of given random parameters, but it can also identify changes in the parameters that define the randomness, and it changes its conduct accordingly. (More details on this and related research can be found in the monograph by Mangel and Clarke.)

People behave differently in situations of randomness than they do in states of uncertainty deriving from lack of clarity. For example, many people are prepared to buy lottery tickets, although the chances of winning are low. Yet they will hesitate before buying a ticket if they do not know how the winner is chosen. A study (published in 2010) on chimpanzees and bonobos (pygmy chimpanzees) by Alexandra Rosati and Brian Hare of Duke University showed that the apes exhibited patterns of behavior similar to that of humans. They could distinguish between situations of lack of knowledge due to randomness with laws of probability and uncertainty not necessarily related to randomness. Moreover, their reactions to these two situations were similar to those of humans, that is, lack of clarity led to a reaction that was more hesitant than the reaction to randomness.

A series of experiments on macaque monkeys and dolphins carried out by David Smith of the State University of New York at Buffalo and David Washborn of Georgia State University (with results published between 1995 and 2003) showed that these two species were aware of the fact that there were things they did not know. The study of the dolphins consisted of training them to react by pressing one of two pedals when they heard a note that was higher or lower in pitch than a certain note, but they also had the opportunity to press a third pedal if the height of the note was unclear to them. Not only did the dolphins press the “not sure” pedal at the right times, their behavior and body language also showed signs of hesitation and lack of decisiveness.

These and many other studies on similar issues show that evolution prepared many species of animals, and certainly humans, to understand and to react intuitively to situations in which they find themselves facing uncertainty and randomness. As we shall see in the next chapters, however, there are aspects of randomness for which they and we are less well adapted.

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.