For the 10 years before I was born, Father’s Day had special meaning to my Dad.

Every Phillies fan—like Dad—knows the significance of Father’s Day of 1964: Jim Bunning pitched a perfect game. It was just the fifth time in modern baseball history that someone could claim such an extraordinary feat: 27 up/27 down.

The idea of perfection takes on a unique and special aura in a game where the cliché “fail two out of three times and you go to Cooperstown” is King. In something as fraught with subjection—Ball or Strike? Hit or error? He did/didn’t beat the throw—as baseball, a perfect game may be the sport’s only universal objective truism. The Perfect Game is—and should be—held in high regard.

Fast forward to 22 days before Father’s Day in 2010. Phillie ace Roy Halladay threw a perfect game. Like Bunning’s, it was historic. Like Bunning’s, it was an incredible feat. Unlike Bunning’s, there was a little bit of “Been there, done that” in the public’s reception. A’s lefty Dallas Braden had just tossed a perfecto 20 days before…which was less than one year after White Sox ace Mark Buerhle pitched perfectly for nine innings…which was a mere five years after The Big Unit put up 27 consecutive Big Oh-fers.

And the ink is still wet on our scorecards for Halladay’s perfect game when just four days later—the same week!—Tiger pitcher Armando Galarraga throws his whatever you want to call it. Technically, it was a one hit shutout. In the court of public opinion, it was a perfect game.

What is going on here? The odds of throwing three perfect games in a month’s worth of contests are approximately 2,000,000 to 1. Yet we have Braden’s, Halladay’s and Galarraga’s box scores staring us in the face.

Has the perfect game become too easy? Is it in danger of losing its luster?

Actually…no. Yes, in the 50 seasons from 1911 to 1960, there were just two perfect games. In the half century from 1961 to 2010, there have been 14, with 11 coming in the last 30 seasons. But a brief statistical analysis will show that maybe the better question is not: why have perfect games been so common of late? It might be: Why were there so few when the Greatest Generation listened to baseball via radio while sitting on their front stoops? Two factors—expansion and on base percentage—go a long way in explaining this.

### Expansion

Major League Baseball’s average on base percentage over the last 90 seasons, from 1921 until 2010 (through May 30), has been .331. While there have been some individual variances year over year, this number has been fairly stable over the seasons.

*Note: My background is as a sportswriter and a scout, not a statistician. I did take a statistics class with an unintelligible professor as part of my MBA work at a university that everyone would recognize. I was lucky to escape with a B-. The stats presented below are back of the envelope. It is my hope that you follow the logic, as the statistical crunching could be tightened- and not effect the conclusion.*

If an average OBP is .331, then the average chance of not getting on base is (1-.331) or .669. The odds of throwing a perfect game can be calculated as .669 times .669 times .669…and so on to the 27th power. This is a minuscule number of .001915%.

Go to your random Nationals-Diamondbacks game and the odds of seeing a perfect game at that particular contest are darn small: .001915% or less than 1/500ths of 1 percent. But just like an infinite number of monkeys pounding on an infinite number of keyboards will produce the complete works of Shakespeare, that figure increases as more and more games are played.

Until 1961, Major League Baseball consisted of 16 teams playing 154 games, giving fans 2,464 different chances of seeing a perfect game. Take that .001915% and multiply it by 2,464 games; there is now a 4.7% chance of a perfect game happening that season.

Expand to 20 teams playing 162 games—like MLB did in 1962—and you know have 3,240 chances to see a perfect game. Odds increase to 6.2%. Expand to 30 teams—for a total of 4,860 chances—and the odds are 9.3%. As small as the odds of a perfect game are, the chance of a perfecto have nearly doubled from 1960 to today merely because of expansion. Even with tiny probabilities, a significant increase in the number of chances that those odds could occur will naturally lead to more occurrences of those odds.

### On Base Percentage (OBP)

Billy Beane was not born in a vacuum.

Kudos to the A’s general manager for preaching on base percentage and the value of taking walks. Moneyball has been quoted ad nauseum.

It may surprise people though that the emphasis on OBP is not cutting edge, but rather going back to the future.

From 1921 to 1941, MLB’s average OBP was .344; it never once fell below .330 for a season. The day after Halladay’s perfect game meanwhile, MLB’s average OBP was .329. Those 15 points might not seem like much, but the average 2010 team has almost twice the odds of being the victim of a perfect game than their 1921-1941 counterparts. The statistical odds of 27 straight outs in a game from a .329 OBP team is .0021%, versus .0011% from a 1921-1941 team.

Combine the pre-World War II emphasis on getting on base with the aforementioned smaller league, and the odds of having a perfect game in a season were a paltry 2.8%. And true to form, just one perfect game was thrown during that time—in 1922 by the immortal Charlie Robertson. (Who?)

On base percentage dipped from 1942 to 1962, to an average of .331. Odds of seeing a perfect game were 4.8% a year, meaning that a 20-year span would likely get one and no more. Just one —Don Larsen’s—did occur during that time.

The biggest year-over-year drop in OBP that baseball has ever seen occurred in 1963. 1962’s OBP was .326; 1963’s plummeted to .309. For the next six seasons, MLB’s average OBP stood at just .308. In 1968, it was so low (.299) that the odds of a perfect game that season were a shockingly high 22%. True to form, Catfish Hunter threw one on May 8th of that year, the third perfect game since Bunning’s aforementioned gem. The surge in perfect games, combined with the steep drop in OBP, is hardly a coincidence.

The dominance of The Year of the Pitcher led to a lowered mound and eventually a designated hitter. True to form, OBPs did rise although never to what they were when Ruth, Gehrig and Greenburg reigned supreme; They averaged .323 from 1969 to 1992 and then rose again to .339 from 1993 to 2000. For all of the emphasis on “Moneyball,” working counts and getting on base, the league average OBP in the decade from 2001 to 2010 is lower (.333) than in the ten year block (1991-2000) prior (.336). A lower OBP means more outs…and increases the odds of ringing up 27 in a row.

In insurance, a “1-in-100 year storm” does not mean that the storm will occur once every 100 years. It means that there is a 1% chance of that storm occurring in that given year. We snicker at three “storms of the century” in a ten-year span, but the actuarial tables support a statement that on the surface seems oxymoronic. Similar statements can be made about perfect games: when the odds of a perfect game are 8% (.335 OBP), 9% (.332 OBP) or 10% (.330 OBP) a year, these alleged “once in a generation” events happen with a lot more frequency.

In conclusion, the odds of a perfect game are still low. We should be surprised at the recent surge in perfectos, but only mildly. Batters get on base less than their pre-expansion counterparts. And a lot more games are played now then when Chicago and St. Louis were “the Western swing.” Let’s salute Dallas Braden, Roy Halladay and yes, Armando Galarraga for their outstanding performances, knowing that despite the eerie timing of their gems, 27 up and 27 down is—and will be—unique.

Jeff Sackmann said...

Great stuff!

Remember that perfect games are also dependent on an error-free defense. That makes perfect games a bit more rare, and it adds to the decline you observe. Here are some rates of RBOE/PA:

1970: .013

1990: .012

2009: .009

Jim said...

Good stuff, but I think it should be noted that the higher OBP in the ‘20s and ‘30s was really just a by-product of higher batting averages of the time, for example in 1925 the league BA was .292 with a league OBP of .354, vs current BA/OBP of .258/.330. The emphasis was high batting average and getting on base with a hit, so I don’t think you can imply the ‘modern’ day philosophy of OBP was in vogue back then.

Even so I guess this shows the percentage chance of a no-hitter has improved much more than the percentage chance of a perfect game.

Marcus said...

I just wanted to write in that your math is a bit off, though you made a good effort for someone without much statistical training!

You’re correct that there is (approximately) a .001915% of a perfect game during any particular game. However, this is probably the lower bound of the actual expectancy. Hitters presumably have much lower OBP’s with the bases empty and pitchers have some amount of skill that skews the statistics so the probability of perfection overall is higher. It might also be helpful to consider the error percentage a particular play (which is also probably at its lowest with the bases empty).

However, my big issue is with your extrapolation of the percentage over a full season. Firstly, 16 teams do play 154 games a year, but each game has two participants, so there are actually 1232 games a season in that scenario. Additionally, the correct way to calculate the percentage of a perfect game is the same as what you used to calculate the probably of a perfect game in the first place.

That is, 1 minus the probability that there is a non-perfect game every game of the season, which is 1 minus .00001915 to the 1232nd power. It comes out to 2.4%, which is nearly half as likely as your original calculation.

Regardless, I think you’re right that perfect games have NOT gotten easier and that 3 over such a short period of time certainly qualifies as something of a miracle.

Brian Cartwright said...

Yes, these days batters reach base on error 30% less often, but many of those 30% are ruled infield singles as opposed to the defense converting them into outs.

Dave Studeman said...

This article was mentioned on NPR:

http://www.npr.org/templates/story/story.php?storyId=127481923

Nice job, Don!

Sean said...

Marcus: There are two teams in each game so two chances for a perfect game.

jp981561 said...

Sean: not exactly, a game cannot end with two perfect games because no one scored….there could only be one chance for perfection per each game

Darren said...

Not just mentioned but also with the introduction of something like “From the amazing site Hardballtimes…” Great that you are getting some recognition.

Dave Studeman said...

jp981561, you’re technically right, but it probably doesn’t impact the overall probability very much. Although you can’t wind up with two perfect games in one, that probability of that happening is so low that it doesn’t materially impact the end result. I don’t think. (I learned long ago to take nothing for granted in probability. Remember the Monty Hall problem…)

stevet said...

“Yes, these days batters reach base on error 30% less often, but many of those 30% are ruled infield singles as opposed to the defense converting them into outs.”

Indeed. The culture of official scoring has distinctly changed over the past few decades. Nowdays, unless an infielder absolutely kicks a grounder, it’s ruled a hit. There’s no doubt that the quality of fielding has improved, but it hasn’t improved as much as the improvement in fielding percentage would imply.