The Physics of the Most Perfect Game

Felix Hernandez is the last major league pitcher to throw a perfect game. (via Keith Allison)

Felix Hernandez is the last major league pitcher to throw a perfect game. (via Keith Allison)

It happens. You’re walking down the street and you suddenly feel the need to straighten your tie, fix your hair, or adjust the collar on your coat. You see a weak reflection of yourself in a window and get to work only to realize the people inside the window have a clear view and are trying hard not to laugh.

Now, a normal person would just be embarrassed and walk away as quickly as possible. However, a nerd like me  would continue to stare at the window because this episode brings to mind an experiment to help me grasp the nature of the universe.

Instead of standing in front of a window, I could shine a flashlight at a piece of glass. Most of the light goes straight through, but a small fraction, about four percent, reflects back toward the flashlight. Which part of the light bounces back? Could you build a flashlight without the bounce-back light? So many questions.

The flashlight’s beam, like everything else in our world, can be broken down into smaller entities. Perhaps you have heard of the fundamental particles of light called “photons.” The beam is composed of about a billion-billion identical photons coming out of the flashlight each second.

Since the photons are identical, there is no way to distinguish in advance which ones will reflect off the glass and which will go through. Therefore, the only workable explanation is statistical. As each photon encounters the glass, it has a one-in-25 chance (four percent) of bouncing back.

This statistical behavior turns out to be fundamentally true of everything in nature and is the basis for the science called “Quantum Mechanics.” However, The Hardball Times is about baseball, not physics. So, let’s move on to understanding a bit about how statistics work.

If a billion photons encounter the glass, then about 40 million photons will reflect. However, if you look at 25 photons hitting the glass, you would most likely see one photon reflect, but you would not be surprised to see zero, or two, or maybe even three reflect. Finally, if you examine a single photon you really can’t predict which it will do at all.

The universe – like baseball – runs on statistics. So, if you look at a stat like on-base percentage (OBP) it might give you a good sense about how likely a batter is to reach base over the course of a season. However, it is less helpful if you try to use it over a given week when a hitter might be on a hot streak or in a slump. It is even less predictive for a single at-bat.

Nonetheless, when it comes to perfect games, it is really the only relevant statistic that is easy to find and apply. Below is a table listing all 23 perfect games in major league history.

PERFECT GAMES, MAJOR LEAGUE HISTORY
Pitcher HOF Date Team Opponent
Lee Richmond 6/12/1880 Worcester Ruby Legs Cleveland Blues
Monte Ward 6/17/1880 Providence Grays Buffalo Bisons
Cy Young  5/5/1904 Boston Pilgrims Philadelphia Athletics
Addie Joss 10/2/1908 Cleveland Naps Chicago White Sox
Charlie Robertson 4/30/1922 Chicago White Sox Detroit Tigers
Don Larsen 10/8/1956 New York Yankees Brooklyn Dodgers
Jim Bunning 6/21/1964 Philadelphia Phillies New York Mets
Sandy Koufax  9/9/1965 Los Angeles Dodgers Chicago Cubs
Catfish Hunter  5/8/1968 Oakland Athletics Minnesota Twins
Len Barker 5/15/1981 Cleveland Indians Toronto Blue Jays
Mike Witt 9/30/1984 California Angels Texas Rangers
Tom Browning 9/16/1988 Cincinnati Reds Los Angeles Dodgers
Dennis Martinez 7/28/1991 Montreal Expos Los Angeles Dodgers
Kenny Rogers 7/28/1994 Texas Rangers California Angels
David Wells 5/17/1998 New York Yankees Minnesota Twins
David Cone 7/18/1999 New York Yankees Montreal Expos
Randy Johnson 5/18/2004 Arizona D-backs Atlanta Braves
Mark Buehrle 7/23/2009 Chicago White Sox Tampa Bay Rays
Dallas Braden  5/9/2010 Oakland Athletics Tampa Bay Rays
Roy Halladay 5/29/2010 Philadelphia Phillies Florida Marlins
Philip Humber 4/21/2012 Chicago White Sox Seattle Mariners
Matt Cain 6/13/2012 San Francisco Giants Houston Astros
Felix Hernandez 8/15/2012 Seattle Mariners Tampa Bay Rays

There are few slouches on this list of pitchers. In fact, more than 25 percent are members of the Hall of Fame and there are few pitchers you haven’t at least heard about. The performance most worthy of note, of course, is Don Larsen’s perfecto in the World Series:

Charlie Robertson’s effort against Detroit included an 0-for-3 day by Ty Cobb. Cobb had a rough start in 1922. On that day in April he was batting only .083 with an .154 OBP. He recovered and finished the campaign with his more typical .401 batting average.

Imagine the misfortune of Ossee Schrecongost of the Philadelphia Athletics, who was blanked during Cy Young’s perfect game. He was then traded to the White Sox in time to be blanked by Addie Joss four years later.  Young’s perfect game was thrown against Hall of Famer Rube Waddell and  Joss pitched his against another Hall of Famer, Ed Walsh.

That’s fun and all, but let’s get back to statistics. We’ll start with something relatively simple like dice. Suppose you hold a single die. What are the chances you won’t throw an ace (one)? Well, you’ll throw an ace about one in six throws, so the odds of throwing an ace are one in six, or the probability of throwing an ace is one-sixth. The chances of not throwing a one then must be one minus one-sixth or five-sixths.

Suppose you are going to throw the die two separate times. What are the chances that you will not throw at least one ace? There are 36 possible combinations. Eleven of them have at least one ace. So the probability of throwing no aces is 36 minus 11, or 25 of the 36 possible outcomes. An easier way to calculate this number is to just take the five-sixths chance for the first throw and multiply by the five-sixth chance for the second throw.

Suppose you intend to throw the die 27 times. What are the chances you never throw an ace? Well, I guess it would be five-sixths times five-sixths times five-sixths…until you do it 27 times. If you care, the probability is 0.73 percent, which means the odds are once in 137 tries.

What does this have to do with perfect games? If you accept that the on-base percentage (OBP) is the probability that a batter will reach base, then the probability that he won’t get on base is one minus the OBP. If no batter reaches base, then you have a perfect game.

So, the probability of throwing a perfect game is roughly the product of one minus the OBP for each batter each time he comes to the plate. The box scores from Baseball-Reference.com include the OBP for each batter at the beginning of the game. However, they only go back to the 1920s.

For the four older games I found box scores (without OBPs) at Baseball-Almanac.com. I then went back to Baseball-Reference.com to find the OBP for each batter for the year. Finally, I could calculate the odds for each perfect game and compare them for sheer entertainment value. The table below is sorted from the most likely perfect game to the least likely, with the calculated odds for each.

PERFECT GAME ODDS
Pitcher Date Team Opponent Odds
Sandy Koufax  9/9/1965 Los Angeles Dodgers Chicago Cubs   1,575
Lee Richmond 6/12/1880 Worcester Ruby Legs Cleveland Blues   2,804
Monte Ward 6/17/1880 Providence Grays Buffalo Bisons   3,005
Philip Humber 4/21/2012 Chicago White Sox Seattle Mariners   4,842
David Wells 5/17/1998 New York Yankees Minnesota Twins   8,369
Len Barker 5/15/1981 Cleveland Indians Toronto Blue Jays  10,703
Cy Young  5/5/1904 Boston Pilgrims Philadelphia A’s  10,723
Jim Bunning 6/21/1964 Philadelphia Phillies New York Mets  14,784
Tom Browning 9/16/1988 Cincinnati Reds LA Dodgers  16,833
Don Larsen 10/8/1956 New York Yankees Brooklyn Dodgers  18,799
Addie Joss 10/2/1908 Cleveland Naps Chicago White Sox  19,510
Mike Witt 9/30/1984 California Angels Texas Rangers  21,622
Catfish Hunter  5/8/1968 Oakland Athletics Minnesota Twins  22,501
Charlie Robertson 4/30/1922 Chicago White Sox Detroit Tigers  29,129
Dallas Braden  5/9/2010 Oakland Athletics Tampa Bay Rays  32,751
David Cone 7/18/1999 New York Yankees Montreal Expos  41,183
Dennis Martinez 7/28/1991 Montreal Expos LA Dodgers  43,433
Roy Halladay 5/29/2010 Philadelphia Phillies Florida Marlins  46,074
Felix Hernandez 8/15/2012 Seattle Mariners Tampa Bay Rays  51,421
Matt Cain 6/13/2012 San Francisco Giants Houston Astros  52,858
Kenny Rogers 7/28/1994 Texas Rangers California Angels  57,651
Randy Johnson 5/18/2004 Arizona D-backs Atlanta Braves  87,708
Mark Buehrle 7/23/2009 Chicago White Sox Tampa Bay Rays 121,275

Sandy Koufax was my childhood idol. I remember listening to Vin Scully call his perfect game in 1965. So, it saddened me greatly to see that he had the most likely perfecto of all. The Cubs, as usual, were not a good team. They finished eighth that year — losing 90 games.

Talk about bad timing. On the night in question, two September call-ups, Byron Browne and Don Young, played their first major league game for the Cubs. In addition Chicago’s pitcher, Bob Hendley, couldn’t hit water if he fell out of a boat. He went hitless in 14 at-bats that year, striking out 10 times. In essence, Koufax was really facing only six major league batters that night. Of the six only Ron Santo, Ernie Banks and Billy Williams had OBPs above .300.

By this methodology, Buehrle’s flawless performance in 2009 was the most difficult to accomplish. You might recall that in 2008 Tampa Bay lost the World Series to the Phillies. In 2009, the Rays they were set on returning to the Fall Classic. However, the Yankees and Red Sox were also playing great ball. On July 23, the Rays were 6.5 games behind in the AL East but had a record of 52-44.

Their line-up that day included Ben Zobrist, Evan Longoria, Carl Crawford and B. J. Upton. Only one batter had an OBP of less than .300; Zobrist boasted a gaudy .413. Of course, since it was in the American League the Rays pitcher didn’t bat.

I hope this table of odds for perfect games will start lots of fun conversations-discussions-arguments and you’ll add your thoughts to the comments below. After all, these sorts of statistical gymnastics are not just part of the National Pastime, but they are a fundamental truth about the behavior of our universe.


David Kagan is a physics professor at CSU Chico, and the self-proclaimed "Einstein of the National Pastime." Visit his website, Major League Physics, and follow him on Twitter @DrBaseballPhD.
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Scott
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Scott

Great read!

I was shocked to see Humber as the 4th most likely…..those 2012 Mariners really couldn’t get on base.

Random thought: is there any evidence that these happen on day games after night games or “getaway” days more than random chance would suggest?

Marc Schneider
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Marc Schneider

Would changes in the strike zone affect the odds? Koufax pitched his perfecto at a time when the strike zone was very large. That presumably affected the chances of no one getting on base. On the other hand, it doesn’t look like there was any disproportionate number of perfect games during the large strikezone ere (which I think was about 1963-1968). There were three during that period but there were also three in 2012 alone.

Will
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Will

I would think the strike zone would be one factor baked in to the respective OBPs.

bucdaddy
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bucdaddy

I was wondering what happened around 1980 that created conditions for so many more perfect games. Maybe just more games in general, with MLB about doubling in size since 1961? “Not as many good hitters” I don’t think is plausible, given the vast expansion of the talent market since then. It’s interesting to me that only Witt’s game (after 1980) happened in September, presumably against a lineup loaded with 40-man roster call-ups. Anybody here seen a perfect game in person, or come close to one? I’ve seen a couple reverse no-hitters (leadoff man or first two batters got hits and… Read more »

Wildcard09
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I was at Fenway in mid-2008 when John Lackey took a no-no into the 9th. That game also happened to be the final nail in the coffin for Sox fans and Manny, when he was booed for his lack of hustle. There was one play earlier in the game in which Manny could have easily got an infield hit on a throw that pulled the first baseman off the bag, but he was still out because he was running less than 50%. And I believe it was also the same day that the Angels traded for Teixeira.

Cliff
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Cliff

I saw the Red Sox pitch one against the Blue Jays during spring training, 14 March 2000–Pedro Martinez (3.0 IP) plus five relievers

Eric the Snail
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Eric the Snail

I noticed you forgot to include Armando Galarraga’s perfect game, which, by the way, happened. I ran the numbers and it came out almost exactly tied with Felix Hernandez at 51,327, so therefore Armando Galarraga was exactly as good a pitcher as Felix Hernandez.

hopbitters
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hopbitters

Good stuff. Man, the Dodgers and Rays came out on the wrong side of that list three times each, with the Rays accumulating those zeroes in a very short span.

The closest I ever came to seeing a perfect game was Brandon Webb’s one-hitter against the Cardinals. September 9, 2006 at the BOB. Even at that, it was several plays away from perfect. The one hit was a Scott Rolen double, but there were also two errors, and a HBP.

Anon Mathematician
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Anon Mathematician

As a preliminary thought, I would imagine that wOBA would be more useful than OBP, since it is a stat which could be usefully adjusted to the ballpark in which the game was pitched. Not likely to make a major difference though except in some really close comparisons (e.g. maybe Cy Young and Len Barker swap places). A more significant thought though is that there are a lot of other *really* subtle factors that go into a perfect game which aren’t captured by OBP. Errors, for example, are not reflected at all (nor should they be!). However, an error by… Read more »

mathEmagician
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mathEmagician

Interesting article. A couple of points worth addressing. In looking at a player’s OBP, it should be adjusted, or regressed, toward the mean (think Stein’s paradox: http://statweb.stanford.edu/~ckirby/brad/other/Article1977.pdf). Players with fewer plate appearances will have less reliable OBPs, so games earlier in the season will not be as accurate as games later, in general. Adjusting to the mean OBP of the given year would provide a more accurate OBP. Second, I think probability of reaching a base also needs to be adjusted by the given pitcher. The probability of getting on base is certainly not the same against all pitchers, as… Read more »

Eric
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Eric

Okay, really geeky question for @mathEmagician: Do we care about Stein’s paradox? We don’t want the estimator with lowest MSE for the vector (OBP1, OBP2, . . . OBP9). We want the best estimator for the quantity (1 – OBP1) * (1 – OBP2) * . . . . (And I have no idea whether the estimator he used here is unbiased, has lowest MSE, whatever. But it sure is the obvious choice.)

mathEmagician
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mathEmagician

This was my fault. Although I read the article in full, I apparently didn’t pay enough attention to all the details before making my remark. I thought Mr. Kagan was using OBP coming into the game, but he clearly states he used OBP for the year. Therefore, regressing toward the league average is much less relevant. My apologies.

I still stand by my second comment, regarding adjustment for the pitcher, however.

Jason Conwell
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Jason Conwell

I’m curious as to how the probabilities of perfect games are reflected in the number of games played. Are we seeing the “expected number of photons reflected?” There are currently 2430 games per season so I’m guessing that there has been something like 200,000 games in history (I don’t have time to do the math). We’ve seen 23 perfect games, so it’s about 1 in 9000. That’s in the ballpark of what would he predicted with the odds here. Especially considering that many of these were great pitchers, we would have to assume that the OBP against these pitchers was… Read more »

Brendan
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Brendan

The Tom Browning part links to Tom Brown.

Was he “Johan Santa”-ed, so to speak?