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

CHAPTER V. THE MATHEMATICS OF RANDOMNESS

43. INTUITION VERSUS THE STATISTICS OF RANDOMNESS

Although evolution did not prepare us to analyze intuitively with logical elements situations of uncertainty, we could assume that we would react correctly to statistical situations. Throughout the whole of the evolutionary process, humans have been exposed to random occurrences. Nevertheless, even in these cases errors related to statistical randomness are repeated again and again; we mentioned some of them in section 39 on the mathematics of predictions and errors. Some of the errors can be explained by evolution itself. We will give a few examples.

The Ayalon Highway in Israel that traverses Tel Aviv and its suburbs is intended to enable vehicles to cross the city quickly. Shortly after a central section of the highway was opened with due pomp and ceremony, the Ayalon River overflowed due to very heavy rain, and the highway was flooded. This led to severe traffic jams, and the CEO of the Ayalon Highway Company was invited to appear on television to explain the reasons for the flooding. His explanation was convincing: To build a highway that would be immune to any possible flooding would be prohibitively expensive. The engineers therefore took a calculated risk and constructed a road with a wide margin of error, so that flooding was expected only once in twenty-five years. It was bad luck, he explained, that this flooding occurred only a short time after the highway was inaugurated, but that was the nature of randomness. He went on to calm the viewers that now they could look forward to a long period of flood-free driving on the highway. Exactly three weeks passed, and the highway was flooded again. The CEO was again invited to appear on television and with a crestfallen face mumbled something about independent and dependent events, without managing to persuade the interviewer that the engineers’ calculations were accurate under the circumstances. The reason for the mistake is clear: in his first broadcast the CEO gave insufficient weight to the most important piece of information, that the highway had just been flooded. If a flood occurs because of extremely heavy downpours, the ground is saturated with water and even light rain may then cause a flood, in other words, the next flooding is not an event independent of the first.

The attitude to the significance of numerical values that the law of probability attributes to various events is not uniform or consistent. Some years ago there was a danger that the Sea of Galilee would flood its shores. The executive responsible for Israel's water sector explained on television that the chance of such flooding was 60 percent and went on to say that a miracle was needed to avoid it. Is an occurrence that has a 40 percent chance of taking place considered a miracle? I doubt it. And indeed, that year a “miracle” occurred, and the Sea of Galilee did not flood. To many doctors, an 80 percent chance of a patient's recovery and a 97 percent chance of recovery may seem similar, but to the patient who understands the law of probability, the difference is huge. A 97 percent chance of recovery means that the treatment is successful in all but a few cases. A 20 percent chance of failure indicates that failure is a systemic possibility.

The attitude to events with very low probability is also inconsistent. On the one hand, people buy lottery tickets although the trouble it takes outweighs the probable winnings. The reason is apparently the positive personal feeling of looking forward to a possible win, even though they know it is unlikely to be realized. On the other hand, the intuitive tendency is to ignore events that have very little chance of being realized. Sometimes this tendency is crucial, particularly in financial, economic, political, and similar matters. This tendency to ignore unlikely events may also be traced to evolutionary sources. In the broad framework of the struggle for survival, the means devoted to facing up to occurrences with a small chance of happening are made at the expense of the major efforts needed in the struggle for survival. For example, if dinosaurs would have developed gills that enabled them to breathe dusty air, they would have survived the meteoric dust that according to the generally accepted explanation engulfed the Earth and resulted in their extinction. On the other hand, a species of dinosaur that would have devoted efforts to developing such gills at the expense of the struggle for day-to-day survival may not have survived and may have become extinct before the meteor collided with Earth. The evolutionary struggle is one of here and now, that is, it takes into account only current conditions and ignores possible future events or events with a low probability of occurring. This fact has filtered down into the way we react to dangers that have a low probability of being realized.

Another error that may be called a mental illusion is related to the interpretation of statistical data, and it too may be traced back to evolutionary origins. As we explained in section 4, identifying patterns is an innate ability. Moreover, it is preferable to err on the side of overidentification. Failure to identify an existing pattern may bear a heavy price, compared with the damage that may be suffered through identifying a nonexistent pattern. The psychologist and expert on decision making Amos Tversky (1937–1996), together with his colleagues Thomas Gilovitch and Robert Vallone, decided to examine the “hot hand” belief in basketball. Every basketball fan knows of this phenomenon. When a player scores a number of baskets in successive throws, he, his coach, the opposing team, the spectators, all feel that he has a “hot hand” and that it is reasonable that he should also try to score in the future. In terms of the law of probability, the hot hand rule says that a number of successful shots at the basket increases the chances that the next throw will also be successful, in contrast with the case in which the same player under the same conditions did not score in his previous attempts at the basket. The situation can be explained, and the usually accepted explanation is the combination of achieving self-confidence with the psychological effects following a run of successes.

Tversky and his colleagues decided to examine the hot hand concept, and over a whole NBA basketball season in the United States they watched one of the most successful teams at that time, the Philadelphia 76ers, and recorded each shot at the basket and monitored the runs of successful shots. They discovered, to the surprise of many, that the hot hand was an illusion, a fallacy. In a random series, a run of successes can also occur without the chances of success in the next attempt increasing. The runs (or “streaks”) of successful shots in the 76ers’ games did not differ from those of a random series. The parameters of the series, that is the percent of successful shots, are likely to change from player to player and from one game to another, but under the same conditions the chances of a successful shot at the basket does not increase after a run of successful shots.

This finding should have immediate implications because a player's hot hand, if it does not exist, has a direct effect on how the coach manages the team during the game. Tversky and his colleagues’ findings met with a mixed reception. They had no effect on the spectators, the players, or the coaches, who continued to believe in the hot hand, and who continue to act accordingly. Opinions in the scientific community are divided. Some accept the findings as they are, and others think that the hot hand phenomenon does exist but is expressed differently. I do not know whether it exists or not, but there is a simple explanation for the illusion: the need to look for and find patterns is deeply embodied in our genes, so much so that events such as a run of successes, or successful shots at the basket, or a few consecutive years of hotter than usual weather, or successive stock-exchange profits, a run that is consistent with the statistics of random events we interpret as valid nonrandom occurrences.