Peering through the Eyepiece of Randomness - The Drunkard's Walk: How Randomness Rules Our Lives - Leonard Mlodinow

The Drunkard's Walk: How Randomness Rules Our Lives - Leonard Mlodinow (2008)

Chapter 1. Peering through the Eyepiece of Randomness

I REMEMBER, as a teenager, watching the yellow flame of the Sabbath candles dancing randomly above the white paraffin cylinders that fueled them. I was too young to think candlelight romantic, but still I found it magical—because of the flickering images created by the fire. They shifted and morphed, grew and waned, all without apparent cause or plan. Surely, I believed, there must be rhyme and reason underlying the flame, some pattern that scientists could predict and explain with their mathematical equations. “Life isn’t like that,” my father told me. “Sometimes things happen that cannot be foreseen.” He told me of the time when, in Buchenwald, the Nazi concentration camp in which he was imprisoned and starving, he stole a loaf of bread from the bakery. The baker had the Gestapo gather everyone who might have committed the crime and line the suspects up. “Who stole the bread?” the baker asked. When no one answered, he told the guards to shoot the suspects one by one until either they were all dead or someone confessed. My father stepped forward to spare the others. He did not try to paint himself in a heroic light but told me that he did it because he expected to be shot either way. Instead of having him killed, though, the baker gave my father a plum job, as his assistant. “A chance event,” my father said. “It had nothing to do with you, but had it happened differently, you would never have been born.” It struck me then that I have Hitler to thank for my existence, for the Germans had killed my father’s wife and two young children, erasing his prior life. And so were it not for the war, my father would never have emigrated to New York, never have met my mother, also a refugee, and never have produced me and my two brothers.

My father rarely spoke of the war. I didn’t realize it then, but years later it dawned on me that whenever he shared his ordeals, it was not so much because he wanted me to know of his experiences but rather because he wanted to impart a larger lesson about life. War is an extreme circumstance, but the role of chance in our lives is not predicated on extremes. The outline of our lives, like the candle’s flame, is continuously coaxed in new directions by a variety of random events that, along with our responses to them, determine our fate. As a result, life is both hard to predict and hard to interpret. Just as, looking at a Rorschach blot, you might see Madonna and I, a duck-billed platypus, the data we encounter in business, law, medicine, sports, the media, or your child’s third-grade report card can be read in many ways. Yet interpreting the role of chance in an event is not like intepreting a Rorschach blot; there are right ways and wrong ways to do it.

We often employ intuitive processes when we make assessments and choices in uncertain situations. Those processes no doubt carried an evolutionary advantage when we had to decide whether a saber-toothed tiger was smiling because it was fat and happy or because it was famished and saw us as its next meal. But the modern world has a different balance, and today those intuitive processes come with drawbacks. When we use our habitual ways of thinking to deal with today’s tigers, we can be led to decisions that are less than optimal or even incongruous. That conclusion comes as no surprise to those who study how the brain processes uncertainty: many studies point to a close connection between the parts of our brain that make assessments of chance situations and those that handle the human characteristic that is often considered our prime source of irrationality—our emotions. Functional magnetic resonance imaging, for example, shows that risk and reward are assessed by parts of the dopaminergic system, a brain-reward circuit important for motivational and emotional processes.1 The images show, too, that the amygdala, which is also linked to our emotional state, especially fear, is activated when we make decisions couched in uncertainty.2

The mechanisms by which people analyze situations involving chance are an intricate product of evolutionary factors, brain structure, personal experience, knowledge, and emotion. In fact, the human response to uncertainty is so complex that sometimes different structures within the brain come to different conclusions and apparently fight it out to determine which one will dominate. For example, if your face swells to five times its normal size three out of every four times you eat shrimp, the “logical” left hemisphere of your brain will attempt to find a pattern. The “intuitive” right hemisphere of your brain, on the other hand, will simply say “avoid shrimp.” At least that’s what researchers found in less painful experimental setups. The game is called probability guessing. In lieu of toying with shrimp and histamine, subjects are shown a series of cards or lights, which can have two colors, say green and red. Things are arranged so that the colors will appear with different probabilities but otherwise without a pattern. For example, red might appear twice as often as green in a sequence like red-red-green-red-green-red-red-green-green-red-red-red, and so on. The task of the subject, after watching for a while, is to predict whether each new member of the sequence will be red or green.

The game has two basic strategies. One is to always guess the color that you notice occurs more frequently. That is the route favored by rats and other nonhuman animals. If you employ this strategy, you are guaranteed a certain degree of success but you are also conceding that you will do no better. For instance, if green shows up 75 percent of the time and you decide to always guess green, you will be correct 75 percent of the time. The other strategy is to “match” your proportion of green and red guesses to the proportion of green and red you observed in the past. If the greens and reds appear in a pattern and you can figure out the pattern, this strategy enables you to guess right every time. But if the colors appear at random, you would be better off sticking with the first strategy. In the case where green randomly appears 75 percent of the time, the second strategy will lead to the correct guess only about 6 times in 10.

Humans usually try to guess the pattern, and in the process we allow ourselves to be outperformed by a rat. But there are people with certain types of post-surgical brain impairment—called a split brain—that precludes the right and left hemispheres of the brain from communicating with each other. If the probability experiment is performed on these patients such that they see the colored light or card with only their left eye and employ only their left hand to signal their predictions, it amounts to an experiment on the right side of the brain. But if the experiment is performed so as to involve only their right eye and right hand, it is an experiment on the left brain. When researchers performed those experiments, they found that—in the same patients—the right hemisphere always chose to guess the more frequent color and the left hemisphere always tried to guess the pattern.3

Making wise assessments and choices in the face of uncertainty is a rare skill. But like any skill, it can be improved with experience. In the pages that follow, I will examine the role of chance in the world around us, the ideas that have been developed over the centuries to help us understand that role, and the factors that often lead us astray. The British philosopher and mathematician Bertrand Russell wrote,

We all start from “naive realism,” i.e., the doctrine that things are what they seem. We think that grass is green, that stones are hard, and that snow is cold. But physics assures us that the greenness of grass, the hardness of stones, and the coldness of snow are not the greenness of grass, the hardness of stones, and the coldness of snow that we know in our own experience, but something very different.4 In what follows we will peer at life through the eyepiece of randomness and see that many of the events of our lives, too, are not quite what they seem but rather something very different.

IN 2002 THE NOBEL COMMITTEE awarded the Nobel Prize in Economics to a scientist named Daniel Kahneman. Economists do all sorts of things these days—they explain why teachers are paid so little, why football teams are worth so much, and why bodily functions help set a limit on the size of hog farms (a hog excretes three to five times as much as a human, so a farm with thousands of hogs on it often produces more waste than the neighboring cities).5Despite all the great research generated by economists, the 2002 Nobel Prize was notable because Kahneman is not an economist. He is a psychologist, and for decades, with the late Amos Tversky, Kahneman studied and clarified the kinds of misperceptions of randomness that fuel many of the common fallacies I will talk about in this book.

The greatest challenge in understanding the role of randomness in life is that although the basic principles of randomness arise from everyday logic, many of the consequences that follow from those principles prove counterintuitive. Kahneman and Tversky’s studies were themselves spurred by a random event. In the mid-1960s, Kahneman, then a junior psychology professor at Hebrew University, agreed to perform a rather unexciting chore: lecturing to a group of Israeli air force flight instructors on the conventional wisdom of behavior modification and its application to the psychology of flight training. Kahneman drove home the point that rewarding positive behavior works but punishing mistakes does not. One of his students interrupted, voicing an opinion that would lead Kahneman to an epiphany and guide his research for decades.6

“I’ve often praised people warmly for beautifully executed maneuvers, and the next time they always do worse,” the flight instructor said. “And I’ve screamed at people for badly executed maneuvers, and by and large the next time they improve. Don’t tell me that reward works and punishment doesn’t work. My experience contradicts it.” The other flight instructors agreed. To Kahneman the flight instructors’ experiences rang true. On the other hand, Kahneman believed in the animal experiments that demonstrated that reward works better than punishment. He ruminated on this apparent paradox. And then it struck him: the screaming preceded the improvement, but contrary to appearances it did not cause it.

How can that be? The answer lies in a phenomenon called regression toward the mean. That is, in any series of random events an extraordinary event is most likely to be followed, due purely to chance, by a more ordinary one. Here is how it works: The student pilots all had a certain personal ability to fly fighter planes. Raising their skill level involved many factors and required extensive practice, so although their skill was slowly improving through flight training, the change wouldn’t be noticeable from one maneuver to the next. Any especially good or especially poor performance was thus mostly a matter of luck. So if a pilot made an exceptionally good landing—one far above his normal level of performance—then the odds would be good that he would perform closer to his norm—that is, worse—the next day. And if his instructor had praised him, it would appear that the praise had done no good. But if a pilot made an exceptionally bad landing—running the plane off the end of the runway and into the vat of corn chowder in the base cafeteria—then the odds would be good that the next day he would perform closer to his norm—that is, better. And if his instructor had a habit of screaming “you clumsy ape” when a student performed poorly, it would appear that his criticism did some good. In this way an apparent pattern would emerge: student performs well, praise does no good; student performs poorly, instructor compares student to lower primate at high volume, student improves. The instructors in Kahneman’s class had concluded from such experiences that their screaming was a powerful educational tool. In reality it made no difference at all.

This error in intuition spurred Kahneman’s thinking. He wondered, are such misconceptions universal? Do we, like the flight instructors, believe that harsh criticism improves our children’s behavior or our employees’ performance? Do we make other misjudgments when faced with uncertainty? Kahneman knew that human beings, by necessity, employ certain strategies to reduce the complexity of tasks of judgment and that intuition about probabilities plays an important part in that process. Will you feel sick after eating that luscious-looking seviche tostada from the street vendor? You don’t consciously recall all the comparable food stands you’ve patronized, count the number of times you’ve spent the following night guzzling Pepto-Bismol, and come up with a numerical estimate. You let your intuition do the work. But research in the 1950s and early ’60s indicated that people’s intuition about randomness fails them in such situations. How widespread, Kahneman wondered, was this misunderstanding of uncertainty? And what are its implications for human decision making? A few years passed, and Kahneman invited a fellow junior professor, Amos Tversky, to give a guest lecture at one of his seminars. Later, at lunch, Kahneman mentioned his developing ideas to Tversky. Over the next thirty years, Tversky and Kahneman found that even among sophisticated subjects, when it came to random processes—whether in military or sports situations, business quandaries, or medical questions—people’s beliefs and intuition very often let them down.

Suppose four publishers have rejected the manuscript for your thriller about love, war, and global warming. Your intuition and the bad feeling in the pit of your stomach might say that the rejections by all those publishing experts mean your manuscript is no good. But is your intuition correct? Is your novel unsellable? We all know from experience that if several tosses of a coin come up heads, it doesn’t mean we are tossing a two-headed coin. Could it be that publishing success is so unpredictable that even if our novel is destined for the best-seller list, numerous publishers could miss the point and send those letters that say thanks but no thanks? One book in the 1950s was rejected by publishers, who responded with such comments as “very dull,” “a dreary record of typical family bickering, petty annoyances and adolescent emotions,” and “even if the work had come to light five years ago, when the subject [World War II] was timely, I don’t see that there would have been a chance for it.” That book, The Diary of a Young Girl by Anne Frank, has sold 30 million copies, making it one of the best-selling books in history. Rejection letters were also sent to Sylvia Plath because “there certainly isn’t enough genuine talent for us to take notice,” to George Orwell for Animal Farm because “it is impossible to sell animal stories in the U.S.,” and to Isaac Bashevis Singer because “it’s Poland and the rich Jews again.” Before he hit it big, Tony Hillerman’s agent dumped him, advising that he should “get rid of all that Indian stuff.”7

Those were not isolated misjudgments. In fact, many books destined for great success had to survive not just rejection, but repeated rejection. For example, few books today are considered to have more obvious and nearly universal appeal than the works of John Grisham, Theodor Geisel (Dr. Seuss), and J. K. Rowling. Yet the manuscripts they wrote before they became famous—all eventually hugely successful—were all repeatedly rejected. John Grisham’s manuscript for A Time to Kill was rejected by twenty-six publishers; his second manuscript, for The Firm, drew interest from publishers only after a bootleg copy circulating in Hollywood drew a $600,000 offer for the movie rights. Dr. Seuss’s first children’s book, And to Think That I Saw It on Mulberry Street, was rejected by twenty-seven publishers. And J. K. Rowling’s first Harry Potter manuscript was rejected by nine.8 Then there is the other side of the coin—the side anyone in the business knows all too well: the many authors who had great potential but never made it, John Grishams who quit after the first twenty rejections or J. K. Rowlings who gave up after the first five. After his many rejections, one such writer, John Kennedy Toole, lost hope of ever getting his novel published and committed suicide. His mother persevered, however, and eleven years later A Confederacy of Dunces was published; it won the Pulitzer Prize for Fiction and has sold nearly 2 million copies.

There exists a vast gulf of randomness and uncertainty between the creation of a great novel—or piece of jewelry or chocolate-chip cookie—and the presence of huge stacks of that novel—or jewelry or bags of cookies—at the front of thousands of retail outlets. That’s why successful people in every field are almost universally members of a certain set—the set of people who don’t give up.

A lot of what happens to us—success in our careers, in our investments, and in our life decisions, both major and minor—is as much the result of random factors as the result of skill, preparedness, and hard work. So the reality that we perceive is not a direct reflection of the people or circumstances that underlie it but is instead an image blurred by the randomizing effects of unforeseeable or fluctuating external forces. That is not to say that ability doesn’t matter—it is one of the factors that increase the chances of success—but the connection between actions and results is not as direct as we might like to believe. Thus our past is not so easy to understand, nor is our future so easy to predict, and in both enterprises we benefit from looking beyond the superficial explanations.

WE HABITUALLY UNDERESTIMATE THE EFFECTS of randomness. Our stockbroker recommends that we invest in the Latin American mutual fund that “beat the pants off the domestic funds” five years running. Our doctor attributes that increase in our triglycerides to our new habit of enjoying a Hostess Ding Dong with milk every morning after dutifully feeding the kids a breakfast of mangoes and nonfat yogurt. We may or may not take our stockbroker’s or doctor’s advice, but few of us question whether he or she has enough data to give it. In the political world, the economic world, the business world—even when careers and millions of dollars are at stake—chance events are often conspicuously misinterpreted as accomplishments or failures.

Hollywood provides a nice illustration. Are the rewards (and punishments) of the Hollywood game deserved, or does luck play a far more important role in box office success (and failure) than people imagine? We all understand that genius doesn’t guarantee success, but it’s seductive to assume that success must come from genius. Yet the idea that no one can know in advance whether a film will hit or miss has been an uncomfortable suspicion in Hollywood at least since the novelist and screenwriter William Goldman enunciated it in his classic 1983 book Adventures in the Screen Trade. In that book, Goldman quoted the former studio executive David Picker as saying, “If I had said yes to all the projects I turned down, and no to all the other ones I took, it would have worked out about the same.”9

That’s not to say that a jittery homemade horror video could become a hit just as easily as, say, Exorcist: The Beginning, which cost an estimated $80 million. Well, actually, that is what happened some years back with The Blair Witch Project: it cost the filmmakers a mere $60,000 but brought in $140 million in domestic box office revenue—more than three times the business of Exorcist. Still, that’s not what Goldman was saying. He was referring only to professionally made Hollywood films with production values good enough to land the film a respectable distributor. And Goldman didn’t deny that there are reasons for a film’s box office performance. But he did say that those reasons are so complex and the path from green light to opening weekend so vulnerable to unforeseeable and uncontrollable influences that educated guesses about an unmade film’s potential aren’t much better than flips of a coin.

Examples of Hollywood’s unpredictability are easy to find. Movie buffs will remember the great expectations the studios had for the megaflops Ishtar (Warren Beatty + Dustin Hoffman + a $55 million budget = $14 million in box office revenue) and Last Action Hero (Arnold Schwarzenegger + $85 million = $50 million). On the other hand, you might recall the grave doubts that executives at Universal Studios had about the young director George Lucas’s film American Graffiti, shot for less than $1 million. Despite their skepticism, it took in $115 million, but still that didn’t stop them from having even graver doubts about Lucas’s next idea. He called the story Adventures of Luke Starkiller as taken from “The Journal of the Whills.” Universal called it unproducible. Ultimately 20th Century Fox made the film, but the studio’s faith in the project went only so far: it paid Lucas just $200,000 to write and direct it; in exchange, Lucas received the sequel and merchandising rights. In the end, Star Wars took in $461 million on a budget of $13 million, and Lucas had himself an empire.

Given the fact that green light decisions are made years before a film is completed and films are subject to many unpredictable factors that arise during those years of production and marketing, not to mention the inscrutable tastes of the audience, Goldman’s theory doesn’t seem at all far-fetched. (It is also one that is supported by much recent economic research.)10 Despite all this, studio executives are not judged by the bread-and-butter management skills that are as essential to the head of the United States Steel Corporation as they are to the head of Paramount Pictures. Instead, they are judged by their ability to pick hits. If Goldman is right, that ability is mere illusion, and in spite of his or her swagger no executive is worth that $25 million contract.

Deciding just how much of an outcome is due to skill and how much to luck is not a no-brainer. Random events often come like the raisins in a box of cereal—in groups, streaks, and clusters. And although Fortune is fair in potentialities, she is not fair in outcomes. That means that if each of 10 Hollywood executives tosses 10 coins, although each has an equal chance of being the winner or the loser, in the end there will be winners and losers. In this example, the chances are 2 out of 3 that at least 1 of the executives will score 8 or more heads or tails.

Imagine that George Lucas makes a new Star Wars film and in one test market decides to perform a crazy experiment. He releases the identical film under two titles: Star Wars: Episode A and Star Wars: Episode B. Each film has its own marketing campaign and distribution schedule, with the corresponding details identical except that the trailers and ads for one film say Episode A and those for the other, Episode B. Now we make a contest out of it. Which film will be more popular? Say we look at the first 20,000 moviegoers and record the film they choose to see (ignoring those die-hard fans who will go to both and then insist there were subtle but meaningful differences between the two). Since the films and their marketing campaigns are identical, we can mathematically model the game this way: Imagine lining up all the viewers in a row and flipping a coin for each viewer in turn. If the coin lands heads up, he or she sees Episode A; if the coin lands tails up, it’s Episode B. Because the coin has an equal chance of coming up either way, you might think that in this experimental box office war each film should be in the lead about half the time. But the mathematics of randomness says otherwise: the most probable number of changes in the lead is 0, and it is 88 times more probable that one of the two films will lead through all 20,000 customers than it is that, say, the lead continuously seesaws.11 The lesson is not that there is no difference between films but that some films will do better than others even if all the films are identical.

Such issues are not discussed in corporate boardrooms, in Hollywood, or elsewhere, and so the typical patterns of randomness—apparent hot or cold streaks or the bunching of data into clusters—are routinely misinterpreted and, worse, acted on as if they represented a new trend.

One of the most high profile examples of anointment and regicide in modern Hollywood was the case of Sherry Lansing, who ran Paramount with great success for many years.12 Under Lansing, Paramount won Best Picture awards for Forrest Gump, Braveheart, and Titanic and posted its two highest-grossing years ever. Then Lansing’s reputation suddenly plunged, and she was dumped after Paramount experienced, as Variety put it, “a long stretch of underperformance at the box office.”13

In mathematical terms there is both a short and a long explanation for Lansing’s fate. First, the short answer. Look at this series of percentages: 11.4, 10.6, 11.3, 7.4, 7.1, 6.7. Notice something? Lansing’s boss, Sumner Redstone, did too, and for him the trend was significant, for those six numbers represented the market share of Paramount’s Motion Picture Group for the final six years of Lansing’s tenure. The trend caused BusinessWeek to speculate that Lansing “may simply no longer have Hollywood’s hot hand.”14 Soon Lansing announced she was leaving, and a few months later a talent manager named Brad Grey was brought on board.

How can a surefire genius lead a company through seven great years and then fail practically overnight? There were plenty of theories explaining Lansing’s early success. While Paramount was doing well, Lansing was praised for making it one of Hollywood’s best-run studios and for her knack for turning conventional stories into $100 million hits. When her fortune changed, the revisionists took over. Her penchant for making successful remakes and sequels became a drawback. Most damning of all, perhaps, was the notion that her failure was due to her “middle-of-the-road tastes.” She was now blamed for green-lighting such box office dogs as Timeline and Lara Croft Tomb Raider: The Cradle of Life. Suddenly the conventional wisdom was that Lansing was risk averse, old-fashioned, and out of touch with the trends. But can she really be blamed for thinking that a Michael Crichton bestseller would be promising movie fodder? And where were all the Lara Croft critics when the first Tomb Raider film took in $131 million in box office revenue?

Even if the theories of Lansing’s shortcomings were plausible, consider how abruptly her demise occurred. Did she become risk averse and out of touch overnight? Because Paramount’s market share plunged that suddenly. One year Lansing was flying high; the next she was a punch line for late-night comedians. Her change of fortune might have been understandable if, like others in Hollywood, she had become depressed over a nasty divorce proceeding, had been charged with embezzlement, or had joined a religious cult. That was not the case. And she certainly hadn’t sustained any damage to her cerebral cortex. The only evidence of Lansing’s newly developed failings that her critics could offer was, in fact, her newly developed failings.

In hindsight it is clear that Lansing was fired because of the industry’s misunderstanding of randomness and not because of her flawed decision making: Paramount’s films for the following year were already in the pipeline when Lansing left the company. So if we want to know roughly how Lansing would have done in some parallel universe in which she remained in her job, all we need to do is look at the data in the year following her departure. With such films as War of the Worlds and The Longest Yard, Paramount had its best summer in a decade and saw its market share rebound to nearly 10 percent. That isn’t merely ironic—it’s again that aspect of randomness called regression toward the mean. A Variety headline on the subject read, “Parting Gifts: Old Regime’s Pics Fuel Paramount Rebound,”15 but one can’t help but think that had Viacom (Paramount’s parent company) had more patience, the headline might have read, “Banner Year Puts Paramount and Lansing’s Career Back on Track.”

Sherry Lansing had good luck at the beginning and bad luck at the end, but it could have been worse. She could have had her bad luck at the beginning. That’s what happened to a Columbia Pictures chief named Mark Canton. Described as box office savvy and enthusiastic shortly after he was hired, he was fired after his first few years produced disappointing box office results. Criticized by one unnamed colleague for being “incapable of distinguishing the winners from the losers” and by another for being “too busy cheerleading,” this disgraced man left in the pipeline when he departed such films as Men in Black ($589 million in worldwide box office revenue), Air Force One($315 million), The Fifth Element ($264 million), Jerry Maguire ($274 million), and Anaconda ($137 million). As Variety put it, Canton’s legacy pictures “hit and hit big.”16

Well, that’s Hollywood, a town where Michael Ovitz works as Disney president for fifteen months and then leaves with a $140 million severance package and where the studio head David Begelman is fired by Columbia Pictures for forgery and embezzlement and then is hired a few years later as CEO of MGM. But as we’ll see in the following chapters, the same sort of misjudgments that plague Hollywood also plague people’s perceptions in all realms of life.

MY OWN EPIPHANY regarding the hidden effects of randomness came in college, when I took a course in probability and began applying its principles to the sports world. That is easy to do because, as in the film business, most accomplishments in sports are easily quantified and the data are readily available. What I discovered was that just as the lessons of persistence, practice, and teamwork that we learn from sports apply equally to all endeavors of life, so do the lessons of randomness. And so I set out to examine a tale of two baseball sluggers, Roger Maris and Mickey Mantle, a tale that bears a lesson for all of us, even those who wouldn’t know a baseball from a Ping-Pong ball.

The year was 1961. I was barely of reading age, but I still recall the faces of Maris and his more popular New York Yankees teammate, Mantle, on the cover of Life magazine. The two baseball players were engaged in a historic race to tie or break Babe Ruth’s beloved 1927 record of 60 home runs in one year. Those were idealistic times when my teacher would say things like “we need more heroes like Babe Ruth,” or “we never had a crooked president.” Because the legend of Babe Ruth was sacred, anyone who might challenge it had better be worthy. Mantle, a courageous perennial slugger who fought on despite bad knees, was the fans’—and the press’s—overwhelming favorite. A good-looking, good-natured fellow, Mantle came across as the kind of all-American boy everyone hoped would set records. Maris, on the other hand, was a gruff, private fellow, an underdog who had never hit more than 39 home runs in a year, much less anywhere near 60. He was seen as a nasty sort, someone who didn’t give interviews and didn’t like kids. They all rooted for Mantle. I liked Maris.

As it turned out, Mantle’s knees got the best of him, and he made it to only 54 home runs. Maris broke Ruth’s record with 61. Over his career, Babe Ruth had hit 50 or more home runs in a season four times and twelve times had hit more than anyone else in the league. Maris never again hit 50 or even 40 and never again led the league. That overall performance fed the resentment. As the years went by, Maris was criticized relentlessly by fans, sportswriters, and sometimes other players. Their verdict: he had crumbled under the pressure of being a champion. Said one famous baseball old-timer, “Maris had no right to break Ruth’s record.”17 That may have been true, but not for the reason the old-timer thought.

Many years later, influenced by that college math course, I would learn to think about Maris’s achievement in a new light. To analyze the Ruth-Mantle race I reread that old Life article and found in it a brief discussion of probability theory18 and how it could be used to predict the result of the Maris-Mantle race. I decided to make my own mathematical model of home run hitting. Here’s how it goes: The result of any particular at bat (that is, an opportunity for success) depends primarily on the player’s ability, of course. But it also depends on the interplay of many other factors: his health; the wind, the sun, or the stadium lights; the quality of the pitches he receives; the game situation; whether he correctly guesses how the pitcher will throw; whether his hand-eye coordination works just perfectly as he takes his swing; whether that brunette he met at the bar kept him up too late or the chili-cheese dog with garlic fries he had for breakfast soured his stomach. If not for all the unpredictable factors, a player would either hit a home run on every at bat or fail to do so. Instead, for each at bat all you can say is that he has a certain probability of hitting a home run and a certain probability of failing to hit one. Over the hundreds of at bats he has each year, those random factors usually average out and result in some typical home run production that increases as the player becomes more skillful and then eventually decreases owing to the same process that etches wrinkles in his handsome face. But sometimes the random factors don’t average out. How often does that happen, and how large is the aberration?

From the player’s yearly statistics you can estimate his probability of hitting a home run at each opportunity—that is, on each trip to the plate.19 In 1960, the year before his record year, Roger Maris hit 1 home run for every 14.7 opportunities (about the same as his home run output averaged over his four prime years). Let’s call this performance normal Maris. You can model the home run hitting skill of normal Maris this way: Imagine a coin that comes up heads on average not 1 time every 2 tosses but 1 time every 14.7. Then flip that coin 1 time for every trip to the plate and award Maris 1 home run every time the coin comes up heads. If you want to match, say, Maris’s 1961 season, you flip the coin once for every home run opportunity he had that year. By that method you can generate a whole series of alternative 1961 seasons in which Maris’s skill level matches the home run totals of normal-Maris. The results of those mock seasons illustrate the range of accomplishment that normal Maris could have expected in 1961 if his talent had not spiked—that is, given only his “normal” home run ability plus the effects of pure luck.

To have actually performed this experiment, I’d have needed a rather odd coin, a rather strong wrist, and a leave of absence from college. In practice the mathematics of randomness allowed me to do the analysis employing equations and a computer. In most of my imaginary 1961 seasons, normal Maris’s home run output was, not surprisingly, in the range that was normal for Maris. Some mock seasons he hit a few more, some a few less. Only rarely did he hit a lot more or a lot less. How frequently did normal Maris’s talent produce Ruthian results?

I had expected normal Maris’s chances of matching Ruth’s record to be roughly equal to Jack Whittaker’s when he plopped down an extra dollar as he bought breakfast biscuits at a convenience store a few years back and ended up winning $314 million in his state Powerball lottery. That’s what a less talented player’s chances would have been. But normal Maris, though not Ruthian, was still far above average at hitting home runs. And so normal Maris’s probability of producing a record output by chance was not microscopic: he matched or broke Ruth’s record about 1 time every 32 seasons. That might not sound like good odds, and you probably wouldn’t have wanted to bet on either Maris or the year 1961 in particular. But those odds lead to a striking conclusion. To see why, let’s now ask a more interesting question. Let’s consider all players with the talent of normal Maris and the entire seventy-year period from Ruth’s record to the start of the “steroid era” (when, because of players’ drug use, home runs became far more common). What are the odds that some player at some time would have matched or broken Ruth’s record by chance alone? Is it reasonable to believe that Maris just happened to be the recipient of the lucky aberrant season?

History shows that in that period there was about 1 player every 3 years with both the talent and the opportunities comparable to those of normal Maris in 1961. When you add it all up, that makes the probability that by chance alone one of those players would have matched or broken Ruth’s record a little greater than 50 percent. In other words, over a period of seventy years a random spike of 60 or more home runs for a player whose production process merits more like 40 home runs is to be expected—a phenomenon something like that occasional loud crackle you hear amid the static in a bad telephone connection. It is also to be expected, of course, that we will deify, or vilify—and certainly endlessly analyze—whoever that “lucky” person turns out to be.

We can never know for certain whether Maris was a far better player in 1961 than in any of the other years he played professional baseball or whether he was merely the beneficiary of good fortune. But detailed analyses of baseball and other sports by scientists as eminent as the late Stephen Jay Gould and the Nobel laureate E. M. Purcell show that coin-tossing models like the one I’ve described match very closely the actual performance of both players and teams, including their hot and cold streaks.20

When we look at extraordinary accomplishments in sports—or elsewhere—we should keep in mind that extraordinary events can happen without extraordinary causes. Random events often look like nonrandom events, and in interpreting human affairs we must take care not to confuse the two. Though it has taken many centuries, scientists have learned to look beyond apparent order and recognize the hidden randomness in both nature and everyday life. In this chapter I’ve presented a few glimpses of those workings. In the following chapters I shall consider the central ideas of randomness within their historical context and describe their relevance with the aim of offering a new perspective on our everyday surroundings and hence a better understanding of the connection between this fundamental aspect of nature and our own experience.