The Canon (10 page)

Through exercises like Birthday Buddies, Nolan's students begin to see the world as both surprisingly predictable and full of surprises. It is a place where small numbers can take on grand airs and seem, on first pass, more meaningful than they are: how could a meager number like 23 possibly perform like 365 without some sort of cosmic motivational speaker prodding it from behind?

It is also a venue large enough for rarities to become regulars, where so many millions of lottery tickets have been sold that ridiculous patterns emerge. A sixty-year-old Australian man buys a Lotto ticket before leaving for vacation, worries that he bought the wrong sort of ticket, and asks a friend back in Sydney to buy another, then frets on returning home that his friend fumbled the request and so decides to spring for a third entry—and ends up with three winning tickets in hand. A woman in Milwaukee, Wisconsin, responds to her husband's hankering for an expensive experimental airplane kit, Sure, honey, go ahead and splurge "when you win the lottery," just as her father had won the state's $2.7 million Megabucks jackpot a dozen years earlier; her husband takes the suggestion seriously and scores a $2.5 million
Megabucks prize himself. Or officials for a large multistate Powerball lottery drawing become suspicious when 110 players scattered among the 29 participating states come in to claim second-place prizes, rather than the 4 or 5 such winners expected from the drawing. But each of the 110 petitioners had guessed 5 out of the 6 Powerball numbers correctly, and each was entitled to anywhere from $100,000 to $500,000 apiece, depending on the initial bet. Behind the startling outbreak of good fortune was a fortune cookie. All the second-place winners had based their choice on the 6 digits they'd seen on the little slip of paper tucked inside a Chinese fortune cookie, a fortune that, like the cellophane-wrapped bill brightener that held it, had been produced in bulk at the Wonton Food factory in New York.

Most of us are not accustomed to a probabilistic mindset, and instead approach life with a personalized blend of sensations, convictions, desires, and intuitions. Our gut is certainly a significant piece of property. The gastrointestinal tract measures about thirty feet from throat to rump and accounts for 10 to 15 percent of one's body weight—but its physical dimensions are nothing compared to its metaphoric value, as the source of our cherished "instincts." We meet new people, we size them up and get a "gut feel" for what they're like, and we contrast them with others in our acquaintance until we find the closest fit. Ah, now we've got them sussed, trussed, and mounted. Now we can safely nap. If our gut instinct happens to clash with logic, probability, or evidence, guess which claimant wins?

Jonathan Koehler of the University of Texas admits that he is not always a popular guest at a wedding. He sits at the ceremony and listens to the giddy couple exchange vows of permanent devotion, passion, and respect. He hears the toasts attesting to the unmistakable rightness of the match, how anybody who knows this man and this woman could tell from the start that the union was "meant to be" and is "like no other," and he thinks, Hmm, I've been to four weddings in the past year. Who's it going to be, then: Zack and Jenny? Sam and Brianna? Brad and Briana? Or Adam and Hermione, now lip-locked so protractedly before me? Which two of these four pairs of besotted newlyweds will end up carrying botulinum-tipped spears into divorce court ten years hence? After all, minor fluctuations notwithstanding, the American divorce rate has been remarkably stable at 50 percent for nearly half a century.

Koehler is friendly and chatty and sometimes shares his musings with other wedding guests. They look at him as though he had belched, or speculated on the correlation between the size of the bride's brassiere and that of the groom's paycheck.

"They find it repugnant to talk statistics at a wedding," he said. "They want to know how I can say such a thing. Why, you don't know anything about this couple! Just look at how happy they are, how deeply in love, how overjoyed their families are. True enough—but I know general frequency statistics. I also know that every couple gets married with kisses and toasts and high hopes, so these details shouldn't affect the probabilities we assign to them. Until you tell me something outside the norm, something diagnostic that has been shown to affect one's probability of divorce—for example, both partners being over the age of thirty-five, which is known to lower the probability of divorce—I'll assume the normal statistical risk applies." Koehler, who has the slight build and dark, floppy hair of Michael J. Fox, insists he's not a "cynical, bitter little man" or a self-satisfied bachelor: to the contrary, he recently got married himself. He's simply accustomed to viewing the world as an extravaganza of sample spaces.

"People don't tend to pay attention to the background information, the sample space," he said. "They take the foreground information without context, and they accept it at face value."

And while full frontal credulity may be the lubricant of matrimony, he said, at other times it helps to look at the big-sky backdrop. More than once Koehler has calmed a jittery passenger seated next to him on an airplane by quoting probabilities. You would have to fly on a commercial airline every day for 18,000 years, he tells them, before your chances of being in a crash would exceed 50 percent. You want to know what 18,000 years looks like? Think "twice as far back as the dawn of agriculture."

Koehler has also examined the errors that people make in deciding how to invest their money. In one study, he and his colleague Molly Mercer showed subjects mockups of advertisements for mutual funds. To the first group they displayed an ad from a small company with a phenomenal track record. It operated only two funds, but each consistently outshone a benchmark market index like Standard & Poor's. Now it was starting up a third fund: Wanna invest? The next set of subjects was treated to an ad from a large mutual fund company, which mentioned that it ran thirty funds and then showed the results of the two funds that "killed" the market index; it, too, was seeking investors for a new fund. Yet another group saw a pitch from the same large company, again attempting to entice investors to a new fund by highlighting the lavish returns on its two star funds, but this time with no reference to the many other, and presumably far less impressive, money sinks in its corporate portfolio.

Koehler and Mercer found that subjects generally were impressed by the small company's results and voiced a willingness to buy into their latest start-up fund. They were equally unimpressed by the big company with thirty funds. "People recognized that, Oh, you're showing me only the best two out of thirty, and they said, Sorry, not interested," said Koehler. But when confronted with ad number three, from the big company that boasts of its two knockouts while omitting any reference to its baseline operations, subjects again fell prey to the lure of the fabulous foreground. They greeted it with the same enthusiasm accorded the small company.

"From a mathematical standpoint, the fund from the investment group that's two-for-two is a much better risk and is much likelier to outperform the market than is that of a group that's two-for-question-mark," said Koehler. "But people often forget to ask, What's the question mark here? They're not thinking about the sample space."

Unfortunately for us poor hayseeds seeking a place to plant our paychecks, real-life advertisements for mutual funds are not legally obliged to divulge their losers and thus they rarely do. Even the advice of "experts" may not enhance our prospects. "We got the same pattern of responses to our ads," Koehler said, "whether we asked undergraduates or professional investors."

Koehler conceded that it's not easy to think about a sample space, the background context, the teeming multitudes beyond the home team in front of you. "We're not hard-wired to think probabilistically," he said. "We're hard-wired to respond to life subjectively, empathetically, and on the fly, which may be a generous impulse in some cases, but at other times it clouds our judgment and is flat-out wrong." One approach he takes to encourage a quantitative mindset is applying it right where subjectivity has the greatest stranglehold on sense: our people skills. He uses exercises like the notorious Linda Problem. Students are given a paragraph describing a hypothetical character named Linda, who is said to be a thirty-year-old American woman who majored in philosophy, graduated with high honors, and has been active in the nuclear freeze and antidiscrimination movements.

Following that tapas of a biography are eight statements, which the readers are asked to rank in order of probability that they apply to Linda. Among them: Linda is a bank teller; Linda is a feminist; Linda is married and has two children; Linda lives in a university town; Linda is a feminist and a bank teller.

Time and again, Koehler said, readers think they know Linda. She's a feminist—that they rank high. And she probably lives in a university
town. The married-with-kids part, who can say, so that gets a listing somewhere in the middle. But a bank teller? That description doesn't sound like Linda at all, and it earns an average ranking way at the bottom of the stack. She could, however, be a feminist
and
a bank teller, couldn't she? Readers assign the composite declaration a higher probability than bank teller alone. "Almost ninety percent of people do this," said Koehler. "They argue, she's definitely not a bank teller, but she could easily be a bank teller and a feminist. At least that's got some of Linda in there. That seems to be the way people think about probability."

There is, of course, a higher probability of Linda being a bank teller than a bank teller and a feminist. In order to be a bank teller and a feminist, she must be a bank teller; and the unconditional probability of one event occurring—in this case, bank tellerdom—is always going to be greater than the conditional conjunction of that event plus a second event—bank tellerdom and a familiarity with the works of Simone de Beauvoir and Gerda Lerner.

Yet even as people accept that Linda might be a feminist bank teller, they feel uncomfortable thinking of Linda's overall prospect of being a bank teller, period. Some might think that to use the job description alone negates, misrepresents, or shortchanges an essential aspect of her being, just as I've felt compelled to qualify my answer whenever people have asked what my father did for a living: he was a machinist at Otis Elevator Company, I say, but he was also an artist who made intricate pen-and-ink drawings, i.e., he was no Archie Bunker. Alternatively, people might be unconsciously fleshing out the statement "Linda is a bank teller" with a clause, "but she is not a feminist," to place it in direct contrast to the statement "Linda is a feminist and a bank teller."

However understandable and folksy maybe the urge to rank the conditional above the unconditional premise in Likely Lines about Linda, it is incorrect, and when Koehler's students realize the error of their weighs, they feel foolish at first, and then eager to try the trick on family and friends, and finally liberated. Where else can they apply their newfound wisdom, their awareness of how important it is to consider background?

Nowhere is the utility of sample-space tracing more obvious than when interpreting the results of a medical test. As a number of studies have revealed, doctors are not always skilled at estimating probabilities or putting a test result in proper context, which means that patients may be sent into paroxysms of anxiety, soul-searching, and planning of funeral choreography unnecessarily, or at least prematurely.

Let's take as an illustrative but purely hypothetical example the following scenario. You're at the doctor's office for routine maintenance, and you happen to notice a sign advertising the monthly special: an AIDS test that is described as "95 percent accurate." You are not in any of the standard high-risk groups for the disease—though you did have crab lice back in college—but as a conscientious citizen and aspiring hypochondriac, you decide to roll up your sleeve and get screened.

A week later, the receptionist from the temp agency who's been filling in for your doctor's phlebotomist calls with grim news: you tested positive. You feel the blood abandon your head and reconvene around your plantar warts. You can't speak. The receptionist mumbles how sorry she is, and how she loved Tom Hanks in
Philadelphia.
How sorry should
you
be, especially since you've never forgiven Hanks for
The Man with One Red Shoe
? The test is "95 percent accurate." You came out positive. Assuming the results weren't caused by a major mechanical screwup like a swapping of test tubes or charts at the laboratory, there's a 95 percent chance you're infected with the AIDS virus, right?

Unbate your breath. Even if it was your vital fluid that yielded the positive result, the real odds are much, much smaller than 95 percent that you are genuinely HIV-positive. In the lively Port Said of the free market, the definition of a test's accuracy can vary depending on the needs and temperament of its parent pharmaceutical company, but in general this figure would mean the following: on the one hand, the test will accurately detect the human immunodeficiency virus in 95 percent of those who have it but will fail to catch 5 percent of those infected; on the other hand, it will correctly rate as negative 95 percent of all noncarriers, but—and here's where your comfort food comes in—it will mistakenly generate a positive result for 5 percent of uninfected patients. Why should you find solace in a puny false-positive figure like 5 percent? Because the potential pool, the sample space, embodied in that figure is formidable. In the United States, HIV infection remains relatively rare, afflicting about 1 in 350 people. Taking a more population-worthy slant on the problem, that means in a random group of 100,000 Americans, some 285 will be HIV-positive, and 99,715 not. Yet if we screened all 100,000 with our AIDS test, what would we expect? The assay would accurately pick up 271 of the 285 viral carriers; but it would slap a fallacious writ of panic on some 4,986 noncarriers. To calculate the odds that a positive result means you are actually infected, you divide the total number of true positives you'd expect in your sample space (271) by the total number of positives overall—false (4,986) and true (271) together. Slice 271 by 5,257, and you end up with a
probability of 5 percent. The gist of that calamitous phone call, then, amounts to the flip figure of your initial fears: there is a 95 percent chance you're virus-
free
.
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