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John Paulos: INNUMERACY: Mathematical Illiteracy and Its Consequences

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John Paulos INNUMERACY: Mathematical Illiteracy and Its Consequences

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Paulos speaks mainly of the dangers of mathematical innumeracy; that is, the common misconceptions of the layperson in regards to numbers, exploring the relationship between math and the human mind. Paulos discusses innumeracy with quirky anecdotes, scenarios and facts, encouraging readers in the end to look at their world in a more quantitative way. Topics include probability and coincidence, the birthday problem, innumeracy in pseudoscience, and statistics and trade-offs in society.

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2 Probability and Coincidence

It is no great wonder if, in the long process of time, while fortune takes her course hither and thither, numerous coincidences should spontaneously occur.

Plutarch

"You're a Capricorn, too. That's so exciting."


A man who travels a lot was concerned about the possibility of a bomb on board his plane. He determined the probability of this, found it to be low but not low enough for him, so now he always travels with a bomb in his suitcase. He reasons that the probability of two bombs being on board would be infinitesimal.

SOME BIRTHDAY VS. A PARTICULAR BIRTHDAY

Sigmund Freud once remarked that there was no such thing as a coincidence. Carl Jung talked about the mysteries of synchronicity. People in general prattle ceaselessly about ironies here and ironies there. Whether we call them coincidences, synchronicities, or ironies, however, these occurrences are much more common than most people realize.

Some representative examples: "Oh, my brother-in-law went to school there, too, and my friend's son cuts the principal's lawn, and my neighbor's daughter knows a girl who once was a cheerleader for the school." – "There've been five instances of the fish idea since this morning when she told me of her fears about his fishing on the open lake. Fish for lunch, the fish motif on Caroline's dress, the…" – Christopher Columbus discovered the New World in 1492 and his fellow Italian Enrico Fermi discovered the new world of the atom in 1942. – "You said you wanted to keep up with him, but later you said you wanted to keep abreast of her. It's clear what's on your mind." -The ratio of the height of the Sears Building in Chicago to the height of the Woolworth Building in New York is the same to four significant digits (1.816 vs. 1816) as the ratio of the mass of a proton to the mass of an electron. -The Reagan-Gorbachev INF treaty was signed on December 8, 1987, exactly seven years after John Lennon was killed.

A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence. If they anticipate someone else's thought, or have a dream that seems to come true, or read that, say, President Kennedy's secretary was named Lincoln while President Lincoln's secretary was named Kennedy, this is considered proof of some wondrous but mysterious harmony that somehow holds in their personal universe. Few experiences are more dispiriting to me than meeting someone who seems intelligent and open to the world but who immediately inquires about my zodiac sign and then begins to note characteristics of my personality consistent with that sign (whatever sign I give them).

The surprising likelihood of coincidence is illustrated by the following well-known result in probability. Since a year has 366 days (if you count February 29), there would have to be 367 people gathered together in order for us to be absolutely certain that at least two people in the group have the same birthday. Why?

Now, what if we were content to be just 50 percent certain of this? How many people would there have to be in a group in order for the probability to be half that at least two people in it have the same birthday? An initial guess might be 183, about half of 365. The surprising answer is that there need be only twenty-three. Stated differently, fully half of the time that twenty-three randomly selected people are gathered together, two or more of them will share a birthday.

For readers unwilling to accept this on faith, here is a brief derivation. By the multiplication principle, the number of ways in which five dates can be chosen (allowing for repetitions) is (365 x 365 x 365 x 365 x 365). Of all these 3655 ways, however, only (365 x 364 x 363 x 362 x 361) are such that no two of the dates are the same; any of the 365 days can be chosen first, any of the remaining 364 can be chosen second, and so on. Thus, by dividing this latter product (365 x 364 x 363 x 362 x 361) by3655, we get the probability that five people chosen at random will have no birthday in common. Now, if we subtract this probability from 1 (or from 100 percent if we're dealing in percentages), we get the complementary probability that at least two of the five people do have a birthday in common. A similar calculation using 23 rather than 5 yields 1/2, or 50 percent, as the probability that at least two of twenty-three people will have a common birthday.

A couple of years ago, someone on the Johnny Carson show was trying to explain this. Johnny Carson didn't believe it, noted that there were about 120 people in the studio audience, and asked how many of them shared his birthday of, say, March 19. No one did, and the guest, who wasn't a mathematician, said something incomprehensible in his defense. What he should have said is that it takes twenty-three people to be 50 percent certain that there is some birthday in common, not any particular birthday such as March 19. It requires a large number of people, 253 to be exact, to be 50 percent certain that someone in the group has March 19 as his or her birthday.

A brief derivation of the last fact: Since the probability of someone's birthday not being March 19 is 364/365, and since birthdays are independent, the probability of two people not having March 19 as a birthday is 364/365 x 364/365. Thus, the probability of N people not having March 19 as a birthday is (364/365)N, which, when N = 253, is approximately 1/2. Hence, the complementary probability that at least one of these 253 people was born on March 19 is also 1/2, or 50 percent.

The moral, again, is that some unlikely event is likely to occur, whereas it's much less likely that a particular one will. Martin Gardner, the mathematics writer, illustrates the distinction between general and specific occurrences by means of a spinner with the twenty-six letters of the alphabet on it. If the spinner is spun one hundred times and the letters recorded, the probability that the word cat or warm will appear is very small, but the probability of some word's appearing is high. Since I brought up the topic of astrology, Gardner 's examples of the first letters of the names of the months and the planets are particularly appropriate. The months-JFMAMJJASOND-give us jason; the planets-MVEMJSUNP-spell sun. Significant? No.

The paradoxical conclusion is that it would be very unlikely for unlikely events not to occur. If you don't specify a predicted event precisely, there are an indeterminate number of ways for an event of that general kind to take place.

Medical quackery and television evangelism will be discussed in the next chapter, but it should be mentioned here that their predictions are usually sufficiently vague so that the probability of some event of the predicted kind occurring is very high; it's the particular predictions that seldom come true. That some nationally famous politician will undergo a sex-change operation, as a newspaper astrologer-psychic recently predicted, is considerably more likely than that New York 's Mayor Koch will. That some viewer will be relieved of his gastric pains just as a television evangelist calls out the symptoms is considerably more likely than that a particular viewer will be. Likewise, insurance policies with broad coverage which compensates for any mishap are apt to be cheaper in the long run than insurance for a particular disease or a particular trip.

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