## THE ODDS ARE YOU'RE INNUMERATE

By JOHN ALLEN PAULUS

**Prior to the release of his pivotal book, Innumeracy, John Allen Paulus published this article in the New York Times.This article provides a good starting point to think about the difficulty innumeracy creates in discussing assessment.**

Innumeracy, the mathematical analogue of functional illiteracy, afflicts far too many literate people, even widely read and articulate men and women who might cringe if words such as ''imply'' and ''infer'' were confused. They generally react without a trace of embarrassment, however, to even the most egregious numerical solecisms. Once I was at a gathering of writers in which much was being made of the difference between ''continually'' and ''continuously.'' Later that evening, as we were watching the news (another measure of how enjoyable the occasion was), the television meteorologist announced that there was a 50 percent chance of rain for Saturday and a 50 percent chance for Sunday as well, and concluded that there was therefore a 100 percent chance of rain that weekend. I grant the mistake was not hilarious, but no one even smiled.

The recent ''Mathematics Report Card'' released by the Educational Testing Service indicates that more than weather reports are at risk. The rampant innumeracy of our high school students and of the educated public in general is appalling, and since this innumeracy can and does lead to muddled personal decisions, misinformed governmental policies and an increased susceptibility to pseudosciences of all kinds, it's not something that can be easily ignored.

I'm not primarily concerned with esoteric mathematics here, only with some feel for numbers and probabilities, some ability to estimate answers to the ubiquitous questions: How many? How likely? With megaton warheads (equivalent in explosive power to a million tons, or two billion pounds, of TNT) and trillion-dollar budgets a reality, people should have a visceral reaction to the difference between a million, a billion and a trillion. (It helps to note that a million seconds takes less than 12 days to tick by, while a billion seconds requires approximately 32 years, and 32,000 years must pass for a trillion seconds to elapse). They needn't be aware of how fast human hair grows, expressed in miles per hour, or of how many basketballs would fit in the Grand Canyon, but they ought to know roughly the population of the United States, the percentage of the world's population that is Chinese, the distance from New York City to Los Angeles, the odds of winning their state lottery and a host of other common magnitudes.

Without a grasp of such basic numbers, just to cite one example, it is impossible to appreciate the silliness of canceling a European trip because of fear of terrorists. In 1985 when, out of the 28 million Americans who traveled abroad, 17 were killed by terrorists, you were almost 25 times as likely to choke to death (one chance in 68,000), about 300 times as likely to die in a car crash (one chance in 5,300) and nearly 2,000 times as likely to die from the effects of smoking (one chance in 800, the equivalent of three fully loaded jumbo jets crashing each and every day of the year).

Still, what do numbers, probabilities and mathematics in general have to do with books and literature? A couple of minor points first. In addition to the usual reasons for innumeracy, avid readers of fiction sometimes have an extra vulnerability to numerical myopia. Their habit of reading about individuals in extreme and dramatic circumstances may exacerbate the natural tendency we all have to give considerably more weight to unlikely but gripping events than we do to more mundane ones. Novels, after all, must be novel, and if, as Tolstoy's chestnut says, all happy families are the same, it's not surprising that the premises of most stories are biased toward the unusual, the special and the rare. (Soft news reports evince this same bias. Willie Horton, the furloughed murderer, seemed to receive more attention than did a decline in the Massachusetts crime rate; likewise, three stranded whales in Alaska may have received more notice than the famine in the Sudan.) Specific instances of quantitative inaccuracy in works of fiction are common, but frequently can be attributed to the characters rather than to the author, or they can be interpreted as having been inserted for effect. These deliberate exaggerations occur in books as dissimilar as Rabelais's ''Gargantua and Pantagruel'' and Gabriel Garcia Marquez's ''One Hundred Years of Solitude,'' and are certainly nothing to get indignant about. My suspicion, however, is that the allusive, metaphoric use of language in literature, so at odds with its more literal use in scientific and mathematical contexts, hides many unintentional clinkers as well.

Interestingly, the much-derided use of coincidence to further a plot has some mathematical justification. Simply put, in many situations coincidences of one sort or another are vastly more common than most people realize. Two people from different parts of the country seated next to each other on a plane, for instance, have about one chance in 100 of sharing an acquaintance and are in fact extremely likely to know people who know each other. They probably won't discover these connections, but if they do, there's no call for the usual exclamations of surprise. Truly amazing would be the complete absence of all such ''amazing'' coincidences.

Although the numerical imprecisions and distortions associated with fictional (and other) writing may be a trivial matter, they're symptomatic of something much more fundamental. The most obvious causes of innumeracy are poor education and ''math anxiety,'' but the deeper sources are prevailing cultural attitudes, in particular misconceptions about the nature of mathematics. These attitudes and misconceptions lead to an intellectual environment that welcomes and even encourages inadequate mathematical education and pride in ignorance (''I hate math.'' ''Hah. Math was always my worst subject''); they also lead, at least in part, to anxiety when quantitative thinking is required.

The proverbial disparagement of the English as ''a nation of shopkeepers'' persists as a belief that a concern with numbers and details numbs one to the big questions, to the grandeur of the natural world. Mathematics is often taken to be mechanical, the work of low-level technicians who will report to the rest of us anything we absolutely must know. Alternatively, mathematics is sometimes thought to have a constraining character that somehow limits our freedom and self-expression. From Wordsworth to present-day New Agers, the romantic idea of communing directly with nature without the distorting aid of any intervening formalisms has also contributed to a vague distaste for mathematics. Sentiments such as these are unfortunately quite prevalent among literary people and ultimately help bring about the abysmal test scores.

Excessive self-centeredness and an aversion to anything deemed technical results as well in a plethora of novels and stories dealing with frequently tiresome variations on midlife crises, divorce and love triangles. Both simple innumeracy and ignorance of higher mathematics (the latter quite an excusable condition) contribute to this limiting of literary possibilities. There are few mainstream American books to compare to ''Labyrinths'' by Jorge Luis Borges, with its obsession with time, reference and infinitely forking alternatives; to Italo Calvino's ''Cosmicomics,'' with its characters constructed from mathematical formulas disporting themselves among the galaxies; or to Raymond Queneau's ''Hundred Trillion Sonnets'' with its combinatorial experiments - all works in which writing is informed by a mathematically alert sensibility. One possible American example is provided, surprisingly enough, by John Updike, who in ''Roger's Version'' knowledgeably vivifies a number of scientific ideas including, unfortunately, the somewhat hokey one of a computer proving God's existence.

There are certainly enough new technical notions to support such a scientifically aware literature. An example that comes immediately to mind is Benoit Mandelbrot's notion of a ''fractal.'' Used to model such disparate physical phenomena as endlessly jagged coastlines, creased and varicose mountain surfaces and the whorls and eddies of turbulent water, fractals might also be employed to represent the fractured, convoluted quality of human consciousness. Likewise, fictional use might be made of the new ideas concerning complexity in computer science. Of course the author who writes such a book could very well see it sink soundlessly into the increasing gap between C. P. Snow's two cultures, a victim of Plato's perennial war between the poetic and the philosophical understanding. Nevertheless, it seems a risk worth taking.

Whether literary types are more prone to innumeracy or not, it's certain that the abstractness of mathematics is a great obstacle for many intelligent people. Such people may readily understand narrative particulars, but strongly resist impersonal generalities. Since numbers, science and such generalities are intimately related, this resistance can lead to an almost willful mathematical and scientific illiteracy. Numbers have appeal for many only if they're associated with them personally - hence part of the attraction of astrology, biorhythms, Tarot cards and the I Ching, all individually customized ''sciences.''

Innumeracy and the attitudes underlying it provide in fact a fertile soil for the growth of pseudoscience. Linda Goodman's ''Star Signs'' and ''Transformation'' by Whitley Strieber are simply two of the latest examples of the many books on astrology, numerology, visitations and the like that fill bookstore shelves (annoyingly often in the philosophy sections) and appeal disproportionately to those who have little interest in or knowledge of numbers, probability or basic science. Others that attract the same people are Budd Hopkins's ''Intruders'' and Mr. Strieber's earlier ''Communion'' (on U.F.O.'s and aliens), the Shirley MacLaine books on reincarnation and psychic communication, J. Z. Knight's ''Ramtha'' books (on channeling and ancient spirits), George Anderson's conversations with the immortals, New Age books on auras, crystal power and chakras, traditional best sellers like Tarot books and the I Ching, and the collection of volumes on the paranormal that Time-Life Books has recently been hawking on television. Even the Harvard-educated physician and author Michael Crichton, who should know better, claims in his book ''Travels'' that mentally bending spoons is no big deal, and that virtually everybody at a party he attended was gently rubbing spoons, thereby causing them (bowls too) to flex like rubber. And if many of the diet books, get-rich-quick books, medical fad books and books on how to calculate the appropriate tip in a restaurant are added, one begins to feel it may be a blessing that so few Americans read the books they buy.

In ''Pseudoscience and Society in Nineteenth-Century America,'' Arthur Wrobel remarks that belief in phrenology, homeopathy and hydropathy was not confined to the poor and the ignorant, but pervaded much of 19th-century literature. Such credulity is not as extensive in contemporary literature, but astrology is one pseudoscience that does seem to engage a big segment of the reading public. Literary allusions to it abound, appearing in everything from Shakespeare to Don DeLillo's ''Libra.'' A 1986 Gallup poll showed that 52 percent of American teen-agers subscribe to it, as does at least 50 percent of the nation's departing First Couple.

Given these figures, it may not be entirely inappropriate to note here that no mechanism through which the alleged zodiacal influences exert themselves has ever been specified by astrologers. Gravity certainly cannot account for these natal influences, since even the gravitational pull of the attending obstetrician is orders of magnitude greater than that of the relevant planet or planets. Nor is there any empirical evidence; top astrologers (as determined by their peers) have failed repeatedly to associate personality profiles with astrological data at a rate higher than that of chance. Neither of these fatal objections to astrology, of course, is likely to carry much weight with literate but innumerate people who don't estimate magnitudes or probabilities, or who are overimpressed by vague coincidences yet unmoved by overwhelming statistical evidence.

Arrayed against fatuous books on stars, the mind and numbers are some excellent expository works on the same topics. These include, among many others, ''A Brief History of Time'' by Stephen Hawking, Rudy Rucker's ''Mind Tools,'' Martin Gardner's many books and essays, James Gleick's ''Chaos,'' ''The Blind Watchmaker'' by Richard Dawkins, ''The Cosmic Code'' by Heinz Pagels, Timothy Ferris's ''Coming of Age in the Milky Way,'' Stephen Jay Gould's books, Douglas Hofstadter's ''Godel, Escher, Bach,'' Carl Sagan's ''Cosmos,'' ''The Loss of Certainty'' by Morris Kline, ''The Mathematical Experience'' by Philip Davis and Reuben Hersh. With a few exceptions, however, such books don't reach a big readership and, because they are difficult, they preach to the converted and inform the informed.

In regard to mathematics at least, something more is needed - books, articles and columns that address a much larger audience and deal not so much with the subject proper as with critical appraisals of the way it is applied and of the misunderstandings and confusions that arise naturally from these applications. Such writings would not change cultural attitudes toward mathematics, of course, but by making its effects concrete and by relating it to matters that are important to all of us, they might help significantly. An overlooked, yet natural, forum for these discussions is the daily newspapers and the general circulation magazines. (If they can carry regular horoscopes, why not entertainingly written articles on the consequences of number numbness?) Perhaps even a regular feature similar to William Safire's On Language column in The New York Times Magazine could muse over the worst innumeracies of the week or month and explain their relevance.

For example, it's often mentioned in stories on drug or AIDS testing that there is a high percentage of false positives. What does this mean? Phrasing the exposition neutrally so as to avoid any factual questions, I'll assume there is a test for cancer that is 98 percent accurate; if someone has cancer, the test will be positive 98 percent of the time, and if one doesn't have it, the test will be negative 98 percent of the time. (Some tests are more reliable, but many, in particular the Pap test for cervical cancer, are considerably less so.) Assume further that 0.5 percent - one out of 200 people - actually have cancer. Now, imagine that you've taken the test and that your doctor somberly informs you that you've tested positive. The question is: How depressed should you be? The surprising answer is that you should be cautiously optimistic.

To see why, let's continue with this digression a bit and imagine that 10,000 tests for cancer are administered. Of these, how many are positive? On the average, 50 of these 10,000 people (0.5 percent of 10,000) will have cancer, and so, since 98 percent of these 50 will test positive, we will have 49 positive tests. Of the 9,950 cancerless people, 2 percent will test positive, for a total of 199 positive tests (2 percent of 9,950 is 199). Thus, of the total of 248 positive tests (199 + 49 - 248), most (199) are false positives, and so the probability that you have cancer when you have tested positive is forty-nine 248ths, or only about 20 percent - and this for a test that was assumed to be 98 percent accurate. To reiterate, if you have cancer, the test will be positive 98 percent of the time, but only 20 percent of those with positive tests will have cancer.

The implications of this little calculation for universal mandatory testing are obvious and should give pause to proponents of such testing, but that is not its point. It is just one of countless vignettes (involving stock scams, sex discrimination, sports records, elections, accounting procedures, lotteries, parapsychological claims, medical and insurance frauds, coincidences, even probabilistic strategies for choosing the ideal mate) that could be constructed to illuminate various quantitative aspects of our public and private lives. If such scenarios and critical commentaries were more widely understood (whether through books, columns or even compact disks), bludgeoning the innumerate into dumb acquiescence by invoking numbers would not be as easy as it sometimes is.

It is distressing that a society and culture that depend so critically on mathematics and its uses should nevertheless seem so indifferent to the innumeracy and general mathematical ignorance of even its brightest citizens. Gauss will never touch us the way Flaubert does, but people should at least recognize his name.

The recent ''Mathematics Report Card'' released by the Educational Testing Service indicates that more than weather reports are at risk. The rampant innumeracy of our high school students and of the educated public in general is appalling, and since this innumeracy can and does lead to muddled personal decisions, misinformed governmental policies and an increased susceptibility to pseudosciences of all kinds, it's not something that can be easily ignored.

I'm not primarily concerned with esoteric mathematics here, only with some feel for numbers and probabilities, some ability to estimate answers to the ubiquitous questions: How many? How likely? With megaton warheads (equivalent in explosive power to a million tons, or two billion pounds, of TNT) and trillion-dollar budgets a reality, people should have a visceral reaction to the difference between a million, a billion and a trillion. (It helps to note that a million seconds takes less than 12 days to tick by, while a billion seconds requires approximately 32 years, and 32,000 years must pass for a trillion seconds to elapse). They needn't be aware of how fast human hair grows, expressed in miles per hour, or of how many basketballs would fit in the Grand Canyon, but they ought to know roughly the population of the United States, the percentage of the world's population that is Chinese, the distance from New York City to Los Angeles, the odds of winning their state lottery and a host of other common magnitudes.

Without a grasp of such basic numbers, just to cite one example, it is impossible to appreciate the silliness of canceling a European trip because of fear of terrorists. In 1985 when, out of the 28 million Americans who traveled abroad, 17 were killed by terrorists, you were almost 25 times as likely to choke to death (one chance in 68,000), about 300 times as likely to die in a car crash (one chance in 5,300) and nearly 2,000 times as likely to die from the effects of smoking (one chance in 800, the equivalent of three fully loaded jumbo jets crashing each and every day of the year).

Still, what do numbers, probabilities and mathematics in general have to do with books and literature? A couple of minor points first. In addition to the usual reasons for innumeracy, avid readers of fiction sometimes have an extra vulnerability to numerical myopia. Their habit of reading about individuals in extreme and dramatic circumstances may exacerbate the natural tendency we all have to give considerably more weight to unlikely but gripping events than we do to more mundane ones. Novels, after all, must be novel, and if, as Tolstoy's chestnut says, all happy families are the same, it's not surprising that the premises of most stories are biased toward the unusual, the special and the rare. (Soft news reports evince this same bias. Willie Horton, the furloughed murderer, seemed to receive more attention than did a decline in the Massachusetts crime rate; likewise, three stranded whales in Alaska may have received more notice than the famine in the Sudan.) Specific instances of quantitative inaccuracy in works of fiction are common, but frequently can be attributed to the characters rather than to the author, or they can be interpreted as having been inserted for effect. These deliberate exaggerations occur in books as dissimilar as Rabelais's ''Gargantua and Pantagruel'' and Gabriel Garcia Marquez's ''One Hundred Years of Solitude,'' and are certainly nothing to get indignant about. My suspicion, however, is that the allusive, metaphoric use of language in literature, so at odds with its more literal use in scientific and mathematical contexts, hides many unintentional clinkers as well.

Interestingly, the much-derided use of coincidence to further a plot has some mathematical justification. Simply put, in many situations coincidences of one sort or another are vastly more common than most people realize. Two people from different parts of the country seated next to each other on a plane, for instance, have about one chance in 100 of sharing an acquaintance and are in fact extremely likely to know people who know each other. They probably won't discover these connections, but if they do, there's no call for the usual exclamations of surprise. Truly amazing would be the complete absence of all such ''amazing'' coincidences.

Although the numerical imprecisions and distortions associated with fictional (and other) writing may be a trivial matter, they're symptomatic of something much more fundamental. The most obvious causes of innumeracy are poor education and ''math anxiety,'' but the deeper sources are prevailing cultural attitudes, in particular misconceptions about the nature of mathematics. These attitudes and misconceptions lead to an intellectual environment that welcomes and even encourages inadequate mathematical education and pride in ignorance (''I hate math.'' ''Hah. Math was always my worst subject''); they also lead, at least in part, to anxiety when quantitative thinking is required.

The proverbial disparagement of the English as ''a nation of shopkeepers'' persists as a belief that a concern with numbers and details numbs one to the big questions, to the grandeur of the natural world. Mathematics is often taken to be mechanical, the work of low-level technicians who will report to the rest of us anything we absolutely must know. Alternatively, mathematics is sometimes thought to have a constraining character that somehow limits our freedom and self-expression. From Wordsworth to present-day New Agers, the romantic idea of communing directly with nature without the distorting aid of any intervening formalisms has also contributed to a vague distaste for mathematics. Sentiments such as these are unfortunately quite prevalent among literary people and ultimately help bring about the abysmal test scores.

Excessive self-centeredness and an aversion to anything deemed technical results as well in a plethora of novels and stories dealing with frequently tiresome variations on midlife crises, divorce and love triangles. Both simple innumeracy and ignorance of higher mathematics (the latter quite an excusable condition) contribute to this limiting of literary possibilities. There are few mainstream American books to compare to ''Labyrinths'' by Jorge Luis Borges, with its obsession with time, reference and infinitely forking alternatives; to Italo Calvino's ''Cosmicomics,'' with its characters constructed from mathematical formulas disporting themselves among the galaxies; or to Raymond Queneau's ''Hundred Trillion Sonnets'' with its combinatorial experiments - all works in which writing is informed by a mathematically alert sensibility. One possible American example is provided, surprisingly enough, by John Updike, who in ''Roger's Version'' knowledgeably vivifies a number of scientific ideas including, unfortunately, the somewhat hokey one of a computer proving God's existence.

There are certainly enough new technical notions to support such a scientifically aware literature. An example that comes immediately to mind is Benoit Mandelbrot's notion of a ''fractal.'' Used to model such disparate physical phenomena as endlessly jagged coastlines, creased and varicose mountain surfaces and the whorls and eddies of turbulent water, fractals might also be employed to represent the fractured, convoluted quality of human consciousness. Likewise, fictional use might be made of the new ideas concerning complexity in computer science. Of course the author who writes such a book could very well see it sink soundlessly into the increasing gap between C. P. Snow's two cultures, a victim of Plato's perennial war between the poetic and the philosophical understanding. Nevertheless, it seems a risk worth taking.

Whether literary types are more prone to innumeracy or not, it's certain that the abstractness of mathematics is a great obstacle for many intelligent people. Such people may readily understand narrative particulars, but strongly resist impersonal generalities. Since numbers, science and such generalities are intimately related, this resistance can lead to an almost willful mathematical and scientific illiteracy. Numbers have appeal for many only if they're associated with them personally - hence part of the attraction of astrology, biorhythms, Tarot cards and the I Ching, all individually customized ''sciences.''

Innumeracy and the attitudes underlying it provide in fact a fertile soil for the growth of pseudoscience. Linda Goodman's ''Star Signs'' and ''Transformation'' by Whitley Strieber are simply two of the latest examples of the many books on astrology, numerology, visitations and the like that fill bookstore shelves (annoyingly often in the philosophy sections) and appeal disproportionately to those who have little interest in or knowledge of numbers, probability or basic science. Others that attract the same people are Budd Hopkins's ''Intruders'' and Mr. Strieber's earlier ''Communion'' (on U.F.O.'s and aliens), the Shirley MacLaine books on reincarnation and psychic communication, J. Z. Knight's ''Ramtha'' books (on channeling and ancient spirits), George Anderson's conversations with the immortals, New Age books on auras, crystal power and chakras, traditional best sellers like Tarot books and the I Ching, and the collection of volumes on the paranormal that Time-Life Books has recently been hawking on television. Even the Harvard-educated physician and author Michael Crichton, who should know better, claims in his book ''Travels'' that mentally bending spoons is no big deal, and that virtually everybody at a party he attended was gently rubbing spoons, thereby causing them (bowls too) to flex like rubber. And if many of the diet books, get-rich-quick books, medical fad books and books on how to calculate the appropriate tip in a restaurant are added, one begins to feel it may be a blessing that so few Americans read the books they buy.

In ''Pseudoscience and Society in Nineteenth-Century America,'' Arthur Wrobel remarks that belief in phrenology, homeopathy and hydropathy was not confined to the poor and the ignorant, but pervaded much of 19th-century literature. Such credulity is not as extensive in contemporary literature, but astrology is one pseudoscience that does seem to engage a big segment of the reading public. Literary allusions to it abound, appearing in everything from Shakespeare to Don DeLillo's ''Libra.'' A 1986 Gallup poll showed that 52 percent of American teen-agers subscribe to it, as does at least 50 percent of the nation's departing First Couple.

Given these figures, it may not be entirely inappropriate to note here that no mechanism through which the alleged zodiacal influences exert themselves has ever been specified by astrologers. Gravity certainly cannot account for these natal influences, since even the gravitational pull of the attending obstetrician is orders of magnitude greater than that of the relevant planet or planets. Nor is there any empirical evidence; top astrologers (as determined by their peers) have failed repeatedly to associate personality profiles with astrological data at a rate higher than that of chance. Neither of these fatal objections to astrology, of course, is likely to carry much weight with literate but innumerate people who don't estimate magnitudes or probabilities, or who are overimpressed by vague coincidences yet unmoved by overwhelming statistical evidence.

Arrayed against fatuous books on stars, the mind and numbers are some excellent expository works on the same topics. These include, among many others, ''A Brief History of Time'' by Stephen Hawking, Rudy Rucker's ''Mind Tools,'' Martin Gardner's many books and essays, James Gleick's ''Chaos,'' ''The Blind Watchmaker'' by Richard Dawkins, ''The Cosmic Code'' by Heinz Pagels, Timothy Ferris's ''Coming of Age in the Milky Way,'' Stephen Jay Gould's books, Douglas Hofstadter's ''Godel, Escher, Bach,'' Carl Sagan's ''Cosmos,'' ''The Loss of Certainty'' by Morris Kline, ''The Mathematical Experience'' by Philip Davis and Reuben Hersh. With a few exceptions, however, such books don't reach a big readership and, because they are difficult, they preach to the converted and inform the informed.

In regard to mathematics at least, something more is needed - books, articles and columns that address a much larger audience and deal not so much with the subject proper as with critical appraisals of the way it is applied and of the misunderstandings and confusions that arise naturally from these applications. Such writings would not change cultural attitudes toward mathematics, of course, but by making its effects concrete and by relating it to matters that are important to all of us, they might help significantly. An overlooked, yet natural, forum for these discussions is the daily newspapers and the general circulation magazines. (If they can carry regular horoscopes, why not entertainingly written articles on the consequences of number numbness?) Perhaps even a regular feature similar to William Safire's On Language column in The New York Times Magazine could muse over the worst innumeracies of the week or month and explain their relevance.

For example, it's often mentioned in stories on drug or AIDS testing that there is a high percentage of false positives. What does this mean? Phrasing the exposition neutrally so as to avoid any factual questions, I'll assume there is a test for cancer that is 98 percent accurate; if someone has cancer, the test will be positive 98 percent of the time, and if one doesn't have it, the test will be negative 98 percent of the time. (Some tests are more reliable, but many, in particular the Pap test for cervical cancer, are considerably less so.) Assume further that 0.5 percent - one out of 200 people - actually have cancer. Now, imagine that you've taken the test and that your doctor somberly informs you that you've tested positive. The question is: How depressed should you be? The surprising answer is that you should be cautiously optimistic.

To see why, let's continue with this digression a bit and imagine that 10,000 tests for cancer are administered. Of these, how many are positive? On the average, 50 of these 10,000 people (0.5 percent of 10,000) will have cancer, and so, since 98 percent of these 50 will test positive, we will have 49 positive tests. Of the 9,950 cancerless people, 2 percent will test positive, for a total of 199 positive tests (2 percent of 9,950 is 199). Thus, of the total of 248 positive tests (199 + 49 - 248), most (199) are false positives, and so the probability that you have cancer when you have tested positive is forty-nine 248ths, or only about 20 percent - and this for a test that was assumed to be 98 percent accurate. To reiterate, if you have cancer, the test will be positive 98 percent of the time, but only 20 percent of those with positive tests will have cancer.

The implications of this little calculation for universal mandatory testing are obvious and should give pause to proponents of such testing, but that is not its point. It is just one of countless vignettes (involving stock scams, sex discrimination, sports records, elections, accounting procedures, lotteries, parapsychological claims, medical and insurance frauds, coincidences, even probabilistic strategies for choosing the ideal mate) that could be constructed to illuminate various quantitative aspects of our public and private lives. If such scenarios and critical commentaries were more widely understood (whether through books, columns or even compact disks), bludgeoning the innumerate into dumb acquiescence by invoking numbers would not be as easy as it sometimes is.

It is distressing that a society and culture that depend so critically on mathematics and its uses should nevertheless seem so indifferent to the innumeracy and general mathematical ignorance of even its brightest citizens. Gauss will never touch us the way Flaubert does, but people should at least recognize his name.

**CITATION**

Paulus, John Allen. “THE ODDS ARE YOU'RE INNUMERATE.” The New York Times, The New York Times, 1 Jan. 1989, www.nytimes.com/1989/01/01/books/the-odds-are-you-re-innumerate.html.