The Odds Of A Perfect Game

A short time after Matt Cain recorded the 27th out of his perfect game on Wednesday night, the interwebs were alive with debate. The topic: Why have perfect games occurred with greater frequency over the last several years, compared to the first 130-plus years of professional baseball?

Political commentator Keith Olbermann is very concerned. He wrote on his baseball blog Baseball Nerd that there are "historical anomalies" in baseball and that "bizarre statistical thunderstorms occur" but "just as these things sometimes happen, they also sometimes indicate a severe skewing of the sport." He went on, "[T]o respond to Matt Cain’s perfect game by simply jumping up and down and buying souvenir merchandise is to miss a bigger picture, one that isn’t exactly clear yet." Olbermann hinted that pitching was at some sort of sinister tipping point.

Jay Jaffe of SI.com had a different take. Jaffe explained that no-hitters and perfect games were more likely now due to four factors: (1) more games per season, and thus an increased number of opportunities for no-hitters and perfect games; (2) major league batting averages are at their lowest level in four decades; (3) higher strikeout rates; and (4) better defense.

Put those factors together and you’ve got an increased chance of a no-hitter on any given night, one that might be aided by a psychological aspect: if a pitcher carries a no-hitter into the later innings, his fielders are more likely to be on their toes, and willing to lay out for a spectacular and risky play such as the one that the Mets’ Mike Baxter made in Santana’s no-hitter. Baxter crashed into the left field wall, injuring his shoulder while robbing Yadier Molina of a hit in the seventh inning; he was forced not only from the game but to the disabled list. In the seventh inning of Cain’s perfecto, Gregor Blanco skidded on the warning track to make a diving catch that prevented Jordan Schafer from getting a hit; fortunately, he was able to walk away unharmed.

For me, Jaffe's explanation makes more sense, as I tend to eschew conspiracy-type theories in favor of facts and evidence. But even if perfect games are occurring with greater frequency -- whatever the reason -- they are still quite rare.

How rare?

After Philip Humber pitched a perfect game on April 21, Harvard University sophomore Andrew Mooney ran the numbers. In the Stats Driven column on Boston.com, Mooney posed the question: "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

In other words, at the beginning of every major-league game, an average pitcher facing a lineup of average hitters has a .000983 percent chance of pitching a perfect game.

But games aren't played by statistically-average pitchers and hitters. If they were, Mooney calculated that we'd have seen only four perfect games in the history of professional baseball, 18 fewer than have actually taken place. He reached that number as follows:

200,304 games played (as of the date of his column) X 2 (number of starting pitchers in each game) = 400,608

400,608 X .00000983 = 3.94 perfect games or 1 perfect game every 34 seasons

Mooney noted the same trend examined by Olbermann and Jaffe. Without delving into the reasons why perfect games were occurring more frequently, he calculated that the chances of four perfect games occurring in a four-year period was 1.77 in 100,000. "Suffice it to say, they're pretty rare," Mooney wrote.

How rare is pretty rare? How do the odds of four perfect games in four years compare to the odds of other events and occurrences?

With a little help from the Harper's Index archive, and some additional internet sleuthing, here's what I found:

• Odds of four perfect games in four years: 1.77 in 100,000 or 1 in 56,497
• Odds that an amateur bowler will bowl a perfect game: 1 in 11,500
• Odds that a professional bowler will bowl a perfect game: 1 in 137
• Odds of amateur golfer hitting a hole-in-one: 1 in 12,750
• Odds of a professional golfer hitting a hole-in-one: 1 in 3,756
• Odds of 137 consecutive professional tennis games without a service break: 1 in 1,250 matches
• Odds in New York City in 1900 of a person dying in a horse accident: 1 in 19,000
• Odds in New York City in 2011 of a person dying in a car accident: 1 in 26,000
• Odds of winning the $540 million super-lottery jackpot in March 2012: 1 in 175,000,000
• Odds that a tsumani will hit the United States Pacific Northwest in the next 30 years: 1 in 10

Yes, perfect games are occurring more frequently but each one is still a rare occurrence.

And each one is a gift of drama, excitement, tension, and, ultimately, exhilaration.