The Canon (9 page)

While simple facts like name spelling are easy to check and correct, it's much trickier to confront your preconceptions and misconceptions and to articulate how or why you conceive of something as you do. Your ideas may be vague. You're not sure where they came from. You feel stupid when you realize you're wrong, and you don't want to admit it, so you say, To hell with it, I'm no good at this, good-bye. Please don't do that. If you realize you might have put those up arrows on the ascending ball, too, or you weren't sure about the seasons, or you thought the lunar phases were the result of Earth's shadow being cast on the moon, rather than the real reason (that half the moon is always lit by the sun,
and half is always dark, and that as the moon makes its month-long revolution around Earth we see different proportions of its light and dark sides), blame it on the brain and its insatiable greed, for picking up everything it comes upon and storing it in the nearest or most logical slot, which may not be right, but so what. That you have to be willing to make mistakes if you're going to get anywhere is true, and also a truism. Less familiar is the fun that you can have by dissecting the source of your misconceptions, and how, by doing so, you'll realize the errors are not stupid, that they have a reasonable or at least humorous provenance. Moreover, once you've recognized your intuitive constructs, you have a chance of amending, remodeling, or blowtorching them as needed, and replacing them with a closer approximation of science's approximate truths, now shining round you like freshly pressed coins.

A
T THE START
of each semester, Deborah Nolan teaches her elementary statistics students a basic, bilateral lesson in life: that it's really hard to look accidental on purpose; and, on the flip side of the same coin, that randomness can look suspiciously rigged. And what better way to prove her point than by flipping coins?

Nolan divides her class of sixty-five or so students into two groups. The members of one group are instructed to take a coin from their purse, pocket, or friendly neighbor, and to flip the coin one hundred times, recording the results of each toss on a sheet of paper. The other students are told to
imagine
tossing a coin one hundred times, and to write down what they think the outcome would be. After signing their work with an identifying mark known only to themselves, the students are to place the spreadsheets of heads and tails face-down on Nolan's desk.

Nolan then leaves the room, and the students start flipping coins and writing, or coining flips and writing. On returning, Nolan glances over the strings of one hundred Hs and Ts and declares each to be either real tossups or faked ones. Nolan is nearly always right, and the students, she said, are "aghast." They think she must have cheated. They think she peeked or had an informant. But she doesn't need to play Harriet the Spy. As it happens, true happenstance bears a distinctive stamp, and until you are familiar with its pattern, you are likely to think it is messier, more haphazard, than it is. Nolan knows what real randomness looks like, and she knows that it often makes people uncomfortable by not looking random
enough.

In the real tossing of a coin, flick after flick, you will find many
stretches of monotony, strings of five heads or seven tails in a row. Now, this is no big deal if you do it long enough and begin to realize that, in the course of one hundred or two hundred flips, clumping happens. Yet when we watch somebody flip a coin in shorter stretches, and especially if we have something riding on the outcome—who gets to choose the vacation destination, for example, or who has to remove the dead opossum from under the porch—we become very dubious when the coin starts repeating itself.
Six
tails? Where did you get that quarter from anyway? a Tom Stoppard play?
^*
Let
me
try.

In their fantasy flippings, the students compensated for their inherent chariness of "too much coincidence" by frequent hopping back and forth, head to tail. In general, the act of jotting down a triplet would set off an alarm bell in the student's head, resulting in a deliberate change of face. "When I look at the fabricated coin tosses, the length of the longest run of heads or tails is way too short," said Nolan. "And overall, the number of switchbacks between heads and tails is way too high." People know there's a fifty-fifty chance for a given outcome with each toss, and they know that, on average, one hundred tosses will yield something close to fifty heads and fifty tails. OK, forty-eight tails, fifty-two heads, I can live with that. But six tails in a row?

"People want to apply the fifty-fifty rule over a very short period of time," said Nolan. "They have a skewed sense of probabilities, and they think the odds of getting multiple heads or tails in a row are much smaller than they are. In fact, the probability of getting four heads or four tails in a row is one in eight, so there's a pretty high chance of it happening." Nolan derived her figure by using the simple multiplication rule that applies to figuring out coin-flipping odds.
^†
You have, of course, a 50 percent chance of tossing a head (or a tail) with each throw—in other words, a probability of 0.5. To calculate the odds of getting two heads in a row, you multiply the two odds together: 0.5 times 0.5, or 0.25—a 25 percent chance that you, the penny pitcher, would see a pair of Lincolns. If you want to ratchet up the number of flips in your probability estimate, just keep multiplying. The prospect of seeing fourheads emerge with four tosses is thus 0.5 multiplied by itself four times, which works out to a one-in-sixteen chance. But because we specified beforehand that we wanted to calculate the odds of seeing four heads
or
four tails, rather than four heads, period, we must add the two probabilities together, and one-in-sixteen plus one-in-sixteen is one in eight.
^*
Granted, the odds of remaining one-sided decrease considerably with each additional toss. The likelihood of flipping six consecutive heads
or
tails is only one in thirty-two, or about 3 percent. This modest potential, though, applies to a single bout of a half-dozen flips. When you're flipping a coin one hundred times, the odds begin to add up, and so, too, do the clusters.

I tried Nolan's coin-tossing exercise myself several times, and over a dozen rounds of one hundred flips each, I never completed a set of one hundred without getting at least one string of six or seven heads or tails in a row, often more than one unbroken sextuplet per set, as well as many quintuplets and quartets. My record for monotony was nine heads in a row, which even now, knowing what I know and assuming a determination to outfox the instructor, I would feel queasy about including in a display of faux flipping.

Until they're schooled in the expansive possibilities of probability theory, Nolan's students regard the notion of randomness as a kind of nervous tic: sorry, sorry, can't stop twitching! Anything beyond this perpetual pinging and ponging, Abe and his monument, and what would you have? A pattern. From a pattern, it's a small step to assuming a point or a portent, and the next thing you know, some poor rabbit is forfeiting its foot to a key chain. "Because many people don't have a real feel for how likely it is for events to happen, they start to attribute hidden meaning to something that's random," said Nolan. "If they see a run of heads or tails beyond a certain length, they begin looking for reasons."

Here we find the basis for superstitiousness, she said. A chance occurrence occurs. Not knowing the odds behind it, we marvel, Now, really, what are the odds? Surely too tiny for chance!

Alan Guth, a physicist at MIT, described an example from his own family of how easily we turn the random into an omen. An uncle of his, who'd lived alone, had been found dead in his home, and a policeman had come to deliver the bad news to Guth's mother. While the officer was there, Guth's sister, who was traveling on business, happened to
call. "My mother and sister were both shocked at the timing of the call, that it coincided with the policeman's visit, and the news of my uncle's death," said Guth. "They thought there had to be something telepathic about it." When Guth heard from his mother of this "miraculous" instance of kin-based telecommunion, he couldn't help but do some quick calculations. As a rule, his sister phoned their mother about once a week. She tended to call either first thing in the morning or in the evening, when she had a free moment and when her mother was likeliest to be around. The policeman had arrived at his mother's house at about 5:00
P.M.
, and, because there were several solemn orders of business to discuss, his visit had lasted more than an hour, possibly two.

All factors considered, Guth said to me, the odds of his sister calling while the policeman was on-site were on par with flipping five heads or tails in a row. "This is not what I would consider a highly improbable event," said Guth. Lucky, yes, given his mother's need for comfort from a loved one, but nothing for which the telepathy option need be considered.

The more one knows about probabilities, the less amazing the most woo-woo coincidences become. My mother told me an amusing story about an acquaintance of hers whose fate, over a six-month period, had seemed linked to her own as though by an idle Pan. The acquaintance was, appropriately for our purposes, an old math professor of hers. Week after week, my parents kept running into him somewhere on Manhattan's sprawling cultural turnpike—an off-Broadway play, a free piano recital, a Bergman movie, the Monet
Water Lilies
room at the Museum of Modern Art. The first few times, my mother and her professor chortled awkwardly over the similarities of their taste. Soon, they were content to nod vaguely from across the room. The coup de graceless came a few months later, in July, and in another country. My parents were strolling along the boulevard St.-Michel on their first trip to Paris, when who should they see but the good professor, sitting at a café. Judging by the way he held his newspaper ostentatiously in front of his face, my mother knew he had spotted them first.

Had my mother been of a superstitious bent, she might have thought the universe was trying to tell her something. ("Your professor hates you!") She is, however, one of the least superstitious people I know, and she understood that (a) those who like Monet like French art; (b) Paris is famous for its world-class collection of French art; (c) "April in Paris" sounds romantic, but "An American in Paris" sounds like July; and (d) an outdoor café is the best place to while away many hours not drinking
a cup of cold espresso, not smoking the lit Gauloise in the ashtray, and not really reading the
Herald-Tribune.

John Littlewood, a renowned mathematician at the University of Cambridge, formalized the apparent intrusion of the supernatural into ordinary life as a kind of natural law, which he called "Littlewood's Law of Miracles." He defined a "miracle" as many people might: a one-in-a-million event to which we accord real significance when it occurs. By his law, such "miracles" arise in anyone's life at an average of once a month. Here's how Littlewood explained it: You are out and about and barraged by the world for some eight hours a day. You see and hear things happening at a rate of maybe one per second, amounting to 30,000 or so "events" a day, or a million per month. The vast majority of events you barely notice, but every so often, from the great stream of happenings, you are treated to a marvel: the pianist at the bar starts playing a song you'd just been thinking of, or you pass the window of a pawnshop and see the heirloom ring that had been stolen from your apartment eighteen months ago. Yes, life is full of miracles, minor, major, middling C. It's called "not being in a persistent vegetative state" and "having a life span longer than a click beetle's."

And because there is nothing more miraculous than birth, Deborah Nolan also likes to wow her new students with the famous birthday game. I'll bet you, she says, that at least two people in this room have the same birthday. The sixty-five people glance around at one another and see nothing close to a year's offering of days represented, and they're dubious. Nolan starts at one end of the classroom, asks the student her birthday, writes it on the blackboard, moves to the next, and jots likewise, and pretty soon, yup, a duplicate emerges. How can that be, the students wonder, with less than 20 percent of 365 on hand to choose from (or 366 if you want to be leap-year sure of it)? First, Nolan reminds them of what they're talking about—not the odds of matching a particular birthday, but of finding a match,
any
match, somewhere in their classroom sample. She then has them think about the problem from the other direction: What are the odds of them not finding a match? That figure, she demonstrates, falls rapidly as they proceed. Each time a new birth date is added to the list, another day is dinged from the possible 365 that could subsequently be cited without a match. Yet each time the next person is about to announce a birthday, the pool the student theoretically will pick from remains what it always was—365. One number is shrinking, in other words, while the other remains the same, and because the odds here are calculated on the basis of comparing (through multiplication and division) the initial fixed set of possible options with an ever diminishing set of permissible ones, the probability of finding no birthday match in a group of sixty-five plunges rapidly to below 1 percent. Of course, the prediction is only a probability, not a guarantee. For all its abstract and counterintuitive texture, however, the statistic proves itself time and again in Nolan's classroom a dexterous gauge of reality.

If you're not looking for such a high degree of confidence, she adds, but are willing to settle for a fifty-fifty probability of finding a shared birthday in a gathering, the necessary number of participants accordingly can be cut to twenty-three. Throw a couple of dozen people together at a cocktail party, in other words, and you have a slightly better than even chance that two of them will be birth-date mates, who, if they discover the fact, will likely exclaim over the coincidence and segue to a discussion of astrology. Or, if their birthday happens to be February 16, and they're talking to me at this imaginary cocktail party, they will hear of the many other date mates who preceded them—Susan the San Francisco photographer, who always brought her golden Labradors on assignment; Frank the Atlanta businessman, who briefly sublet my apartment and whooped it up at the neighborhood tiki bar; Michelle, my brother's girlfriend; and, first but ever least, Robbie, a high school boyfriend of mine, who was cute and smart and studiously mean. Maybe it was his rising sign, or something his poor mother ate.