John Paulos - INNUMERACY - Mathematical Illiteracy and Its Consequences

Even people who have less restrictive filters and a good feel for numbers will note an increasingly large number of coincidences, due in large measure to the number and complexity of man-made conventions. Primitive man, in noticing the relatively few natural coincidences in his environment, slowly developed the raw observational data out of which science evolved. The natural world, however, does not offer immediate evidence for many such coincidences on its surface (no calendars, maps, directories, or even names). But in recent years the plethora of names and dates and addresses and organizations in a complicated world appears to have triggered many people's inborn tendency to note coincidence and improbability, leading them to postulate connections and forces where there are none, where there is only coincidence.

Our innate desire for meaning and pattern can lead us astray if we don't remind ourselves of the ubiquity of coincidence, an ubiquity which is the consequence of our tendency to filter out the banal and impersonal, of our increasingly convoluted world, and, as some of the earlier examples showed, of the unexpected frequency of many kinds of coincidence. Belief in the necessary or even probable significance of coincidences is a psychological remnant of our simpler past. It constitutes a kind of psychological illusion to which innumerate people are particularly prone.

The tendency to attribute meaning to phenomena governed only by chance is ubiquitous. A good example is provided by regression to the mean, the tendency for an extreme value of a random quantity whose values cluster around an average to be followed by a value closer to the average or mean. Very intelligent people can be expected to have intelligent offspring, but in general the offspring will not be as intelligent as the parents. A similar tendency toward the average or mean holds for the children of very short parents, who are likely to be short, but not as short as their parents. If I throw twenty darts at a target and manage to hit the bull's-eye eighteen times, the next time I throw twenty darts, I probably won't do as well.

This phenomenon leads to nonsense when people attribute the regression to the mean as due to some particular scientific law, rather than to the natural behavior of any random quantity. If a beginning pilot makes a very good landing, it's likely that his next one will not be as impressive. Likewise, if his landing is very bumpy, then, by chance alone, his next one will likely be better. Psychologists Amos Tversky and Daniel Kahneman studied one such situation in which, after good landings, pilots were praised, whereas, after bumpy landings, they were berated. The flight instructors mistakenly attributed the pilots' deterioration to their praise of them, and likewise the pilots' improvement to their criticism; both, however, were simply regressions to the more likely mean performance. Because this dynamic is quite general, Tversky and Kahneman write, "behavior is most likely to improve after punishment and to deteriorate after reward. Consequently, the human condition is such that… one is most often rewarded for punishing others, and most often punished for rewarding them." It's not necessarily the human condition, I would hope, but a remediable innumeracy which results in this unfortunate tendency.

The sequel to a great movie is usually not as good as the original. The reason may not be the greed of the movie industry in cashing in on the first film's popularity, but simply another instance of regression to the mean. A great season by a baseball player in his prime will likely be followed by a less impressive season. The same can be said of the novel after the best-seller, the album that follows the gold record, or the proverbial sophomore jinx. Regression to the mean is a widespread phenomenon, with instances just about everywhere you look. As mentioned in Chapter 2, however, it should be carefully distinguished from the gambler's fallacy, to which it bears a superficial resemblance.

Though chance fluctuations play a very large role in the price of a stock or even of the market in general, especially in the short term, the price of a stock is not a completely random walk, with a constant probability (P) of going up and a complementary probability (1-P) of going down, independent of its past performance. There is some truth to so-called fundamental analysis, which looks to the economic factors underlying a stock's value. Given that there is some rough economic estimate of a stock's value, regression to the mean can sometimes be used to justify a kind of contrarian strategy. Buy those stocks whose performance has been relatively lackluster for the previous couple of years, since they're more likely to regress to their mean and increase in price than are stocks which have performed better than their economic fundamentals would suggest and are therefore likely to regress to their mean and decline in price. A number of studies support this schematic strategy.

Judy is thirty-three, unmarried, and quite assertive. A magna cum laude graduate, she majored in political science in college and was deeply involved in campus social affairs, especially in antidiscrimination and anti-nuclear issues. Which statement is more probable?

(a) Judy works as a bank teller.

(b) Judy works as a bank teller and is active in the feminist movement.

The answer, surprising to some, is that (a) is more probable than (b), since a single statement is always more probable than a conjunction of two statements. That I will get heads when I flip this coin is more probable than that I will get heads when I flip this coin and get a 6 when I roll that die. If we have no direct evidence or theoretical support for a story, we find that detail and vividness vary inversely with likelihood; the more vivid details there are to a story, the less likely the story is to be true.

Getting back to Judy and her job at the bank, psychologically what may happen is that the preamble causes people to confuse the conjunction of statements of alternative (b) ("She's a teller and she's a feminist") with the conditional statement ("Given that she's a teller, she's probably also a feminist"), and this latter statement seems more probable than alternative (a). But this, of course, is not what (b) says.

Psychologists Tversky and Kahneman attribute the appeal of answer (b) to the way people come to probability judgments in mundane situations. Rather than trying to decompose an event into all its possible outcomes and then counting up the ones that share the characteristic in question, they form representative mental models of the situation, in this case of someone like Judy, and come to their conclusion by comparison with these models. Thus, it seems to many people that answer (b) is more representative of someone with Judy's background than is answer (a).

Many of the counter-intuitive results cited in this book are psychological tricks similar to the above, which can induce temporary innumeracy in even the most numerate. In their fascinating book Judgement under Uncertainty, Tversky and Kahneman describe a different variety of the seemingly irrational innumeracy that characterizes many of our most critical decisions. They ask people the following question: Imagine you are a general surrounded by an overwhelming enemy force which will wipe out your 600-man army unless you take one of two available escape routes. Your intelligence officers explain that if you take the first route you will save 200 soldiers, whereas if you take the second route the probability is 1/ 3that all 600 will make it, and 2/ 3that none will. Which route do you take?

Most people (three out of four) choose the first route, since 200 lives can definitely be saved that way, whereas the probability is 2/ 3that the second route will result in even more deaths.