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

CHAPTER V. THE MATHEMATICS OF RANDOMNESS

Can birds calculate probabilities? • How can you calculate the chances of winning an unfinished game? • Is it worthwhile to believe in God? • Why did the municipality of Amsterdam almost go bankrupt? • Who murdered Mrs. Simpson? • Why didn't dinosaurs develop gills against dust? • What are the chances of the Ayalon Highway being flooded? • Is there a “hot hand” in basketball?

35. EVOLUTION AND RANDOMNESS IN THE ANIMAL WORLD

The title of this chapter does not refer to the random part of the process of evolution but to the question that has been with us throughout this book, and that is, to what extent did evolution prepare us to analyze intuitively and understand situations in which randomness plays a part? The question is reasonable. Indeed, uncertainty and randomness appear frequently in nature and were part of the environment of species in the evolutionary struggle. We may assume, therefore, that the evolutionary competition gave rise to the development of intuition with respect to uncertainty.

First, however, we will examine more closely the difference between randomness and uncertainty. A situation of uncertainty is one in which we do not know what the result of an occurrence will be and we do not know the circumstances prevailing at the time of the occurrence. Randomness occurs when we do not know for sure what will occur, but we do know that what happens is controlled by a process with given probabilities. For example, when we throw a die with six faces, we do not know in advance which face will be uppermost, but we do know that each of the six faces has the same probability of being the one. The same applies to tossing a coin, with each of the sides, “heads” or “tails,” having an equal chance of falling uppermost. The probabilities do not have to be equal, but the process will still be random. Thus, if four sides of a cube are blue and the other two sides are red, in a random throw of the cube the chances of a blue side appearing uppermost will be two-thirds, and of a red side, one-third. The probabilities are not the same, but the process is a random one. In contrast, in a situation of general uncertainty the result may not be determined by a probability process. For example, a committee is about to make a decision, and we have no idea of the procedure through which it reaches its decisions. That is a situation of uncertainty where, usually, we cannot assign probabilities to the outcome of the decision-making process.

One way of examining whether the relation to uncertainty in general and randomness in particular are rooted in evolution is to study reactions to such situations in the animal world. We used the same approach with the relation to arithmetic in section 2. The result with regard to randomness is very clear: animals succeed in discerning random states. Moreover, they sometimes use randomness to improve their situation. They also manage to recognize situations of uncertainty unrelated to randomness. Furthermore, they are sometimes aware of the fact that there is something they do not know, and then their behavior is similar to that of humans in similar situations. The conclusion to be drawn is that the relation to randomness and uncertainty derives from evolution. We will describe briefly a number of experiments from among the many that confirm this conclusion.

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.