Leonard Mlodinow is an author and a scientist, and recently, he wrote a book. The Drunkard’s Walk is about how randomness rules our lives. We decided to ask him to write something for us about sports and randomness. He decided to take on the 61-homer season of Roger Maris, and attempt to prove that it was a fluke. His words after the jump.

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. 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. As the years went by, Maris was criticized by fans, sportswriters, and sometimes other players. Many concluded that 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.€ But maybe all that psychological analysis was missing the point: could Maris have simply been a very good home run hitter who got lucky? That is, might his record-breaking year have been, simply, a statistical fluctuation?

To analyze the Ruth-Mantle race I decided to make a 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. 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 pure luck.

To have actually performed this experiment, you’d need a rather odd coin, and a rather strong wrist. In practice the mathematics of randomness allows you 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?

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 number of 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 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. 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.