Chicago White Sox lefthander Philip Humber briefly seized the national sporting spotlight on Saturday, throwing the 21st perfect game in Major League Baseball history in a 4-0 victory over the Seattle Mariners.
As of Sunday, there have been 200,304 MLB contests played and the league has seen just about every feat accomplished dozens, if not hundreds of times, making the perfect game that much more of a special achievement. It’s also something that seems to happen at random; most of the greatest pitchers in baseball history have never thrown one, and two of the last three players to do so had fewer than 20 career wins at the time.
According to the MLB rule book, a perfect game is defined as “when a pitcher (or pitchers) retires each batter on the opposing team during the entire course of a game, which consists of at least nine innings. In a perfect game, no batter reaches any base during the course of the game.” Assuming a typical nine inning game, how difficult is it to prevent 27 consecutive batters from reaching base?
Using records from baseball-reference.com, I obtained the average on-base percentage in major league history, which, dating back to the 1876 season, is .326. However, on-base percentage doesn’t factor in another way a perfect game can be broken up: an error committed by a fielder.
From the historical record, I found that, all time, there have been 497,649 errors recorded in major league play, in 15,341,862 total plate appearances. However, of that error total, some fraction includes errors committed following a hit, when runners are already on base, such as an overthrow of the cutoff man that allows a runner to advance a base. These errors could not occur in a perfect game, as there cannot be any baserunners to begin with. I estimate the proportion of such errors to be one-third.
So, if two-thirds of errors result in a baserunner instead of a routine out, I should also multiply the complete error total by two-thirds to determine the number that might disrupt a perfect game. This amounts to an error rate of 2.16 per 100 plate appearances (.0216).
When I add this modified error rate to on-base percentage, the resulting number—.348—is the probability that the average hitter reaches base, or in other words, breaks up a perfect game. Conversely, the probability that a player does not reach base is 1 - .348 = .652. As an aside, I am treating the number of dropped third strikes and catcher’s interference calls as negligible, having no noticeable effect on the aforementioned probability.
Now, to find the likelihood of a perfect game, given a historically average pitcher facing a lineup of nine historically average hitters, I multiply .652 by itself 27 times. This gives me the probability that 27 consecutive hitters will not reach base against an average pitcher.
(.652)^27 = 0.00000983
As mentioned, there have been 200,304 games played in major league history, at the start of which two pitchers each have a chance to throw a perfect game. This brings us to 400,608 perfect game opportunities. Multiplying this number by the probability of the average perfect game gives me the number of perfect games that we would expect to have been thrown in history, based simply on the odds of their occurrence.
0.00000983 x 400,608 = 3.94 ≈ 4.00 perfect games, or one every 34 seasons
From these calculations, we can see that there have actually been more than five times as many perfect games (21) as the probabilities dictate. In fact, for the expected number of perfect games to equal 21, the historical average on-base percentage would have to be 0.284, or roughly the rate at which hitters reached base in 1883, that noted year of offensive explosion. If you need any more context, that’s lower than what we’d expect hitters to post against a league in which every pitcher was Greg Maddux or Randy Johnson.
Curiously, there has been a relative eruption of perfect games in the last decade. In the last four years alone, four pitchers—Mark Buerhle, Dallas Braden, Roy Halladay, and Humber—have achieved perfection, and one other pitcher, Armando Galarraga of the Tigers, was one blown call at first base from making it five. Given the odds listed above, the chances of four perfect games occurring in this four-year span (15,046 games) are 1.77 in 100,000.
Suffice it to say, they're pretty rare. A major league baseball pitcher has better of odds of dying in a lightning strike than tossing a perfect game. That’s all the more reason to appreciate Humber’s big day, though I don’t anticipate we’ll hear of him striking twice anytime soon.
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Stats Driven features a closer look at statistical analysis, sports strategy and trends within Boston sports. Andrew Mooney, a student at Harvard College and an active member of the Harvard College Sports Analysis Collective, is the primary contributor. Email him at firstname.lastname@example.org and follow him on Twitter at @mooneyar.