What moment defines a perfect game? Is it the clinching pitch, the fist-pump, and the dogpile? Is it the camera shot of the players in the dugout avoiding the starter like he’s got bad news? What about the sensational defensive play that typifies the way perfect games are not simply individual performances? No. The defining moment of any would-be perfect game is the moment it dawns on you. Say, you think to yourself as you glance down at your scorecard, how many hits has he given up? No walks, either, huh?

From that moment on, the narrative of the game changes. It’s a thing to be protected, a delicate embryo with a nine-inning gestation period. By the time it reaches its third trimester, only the most hardened hometown supporters are rooting against it. Most of them, though, don’t last. They become stillborn ideas of a nearly impossible feat—a feat so rare you couldn’t even field an entire roster out of players who had accomplished it.

At least, that’s how it used to be. In this yet-young season, there have for all intents and purposes been three perfect games. This is a mind-boggling unlikely occurrence that demands explanation. It’s like buying a Honus Wagner T206 card for over a million dollars and then finding one, near-mint, at the rummage sale down the block. You’d feel excited, sure, but more than a little cheapened and confused.

When Dallas Braden threw a perfect game, a whole lot of fans thought, “him?”  When Roy Halladay did it just a few weeks later, it made more sense, but it was also suspicious; two perfect games had only occurred in the same year once before, and that was when Rutherford B. Hayes was president. When Armando Galarraga retired every batter he faced a matter of days later, people were as dumbfounded as they were enraged at the call that kept the game out of the record books.

Modeling the Mystical

For some reason, I always thought it was better when perfect games happened during the day. It’s an old-timey kind of thing to have happened during a baseball game, its recent resurgence notwithstanding. The image of a perfect game commands images of hand-operated scoreboards populated by zeroes. It is necessary, of course, that any pitcher who throws a perfect game be dominant, but that dominance is often casual and understated. The calmness and composure of the pitcher belie the incredible feat he performs. So perfect games are as close to mystical events as baseball gets during the regular season.

Mystical events are difficult to fathom, and attempting to understand them can often lead to simple reductionism. But without a model, it may never be possible to wrap our minds around them. So here’s a simple model of a perfect game.

A perfect game is a series of nine consecutive perfect innings thrown by the same starter in the same game.

This is not a completely accurate model of a perfect game. We have ignored the possibility of a tie or extra innings, both of which could confound this model. Nevertheless, the nice thing about a simple model is that we can learn things from it.

Let’s start simple. What percentage of starters’ innings are perfect? When we ask our resident Jane Austen scholar, he gives us an impressive data dump, which we can then render graphically.

The chart above shows the percentage of perfect starter innings divided by total starter innings, going back to 1974. That is a period over which runs per game have increased at a relatively steady pace. It’s also a period over which the percentage of innings thrown by starters has steadily decreased. And yet, the ratio of perfect starter innings to total starter innings has remained mostly flat. If anything, it has declined very slowly, from about 40 percent in 1974 to about 38 percent last season.

Let’s use the higher of those numbers (40 percent, because I like easy math) and see if we can’t figure out the likelihood of a perfect game in a given year. Let’s assume that this 40 percent probability is constant across all pitchers and all innings, and that the probabilities are independent from one inning to another. Now, I know these are patently false assumptions, but the distortive effects shouldn’t be too bad, at least not orders of magnitude so. After all, the good pitchers ought to mostly cancel out the bad, and few managers have the guts to take the ball from a guy with a perfect game going. Again, we’re going for simple math here.

As we noted above, we need nine consecutive perfect innings to get a perfect game. So we take (.40)^9, multiplying the probability by itself nine times, and we get a very small number: 0.00026. That would be the probability of any of our identical starters throwing a perfect game on an individual night. But each team plays 162 games, and there are 30 teams, and if we multiply that out we get 1.3, suggesting we should expect more than one perfect game thrown per year! Of course, that’s not at all what we have seen:

 This chart shows the number of perfect games (blue) and no-hitters (red) each year over the same period as above. Recently, perfect games have tended to happen once every few years. So what gives?

For one thing, our model is simple, but it is also very sensitive to inputs. That nine exponent—you know, the little hitch about a baseball game being a rather drawn-out affair—means that if we change the probability, we can drastically change the result. What if, instead of 40 percent, we use the more recent 38 percent figure (simplicity be damned and dust off that calculator)? Well then the probability is cut nearly in half, and the expected number of perfect games per year falls to 0.8.

Oh, Great, It’s Bad Benny Tonight

But it gets more complicated still if we let the probability of throwing a perfect inning vary over the course of a start. Let’s take two pitchers, Chaz Consistency and Benny Badinning
































Notice that both Chaz and Benny have the same average probability of having a perfect inning. If the entire league were made up of Chaz-clones, we would expect just over one perfect game per year. If, on the other hand, it was nothing but Bennies, we’d expect just under one perfect game per year (it’s approximately 1.0 versus 0.9).

And it’s much more likely that the probabilities vary wildly from inning to inning. Why? Because managers lump their best hitters together in the lineup, meaning that one third of the lineup is the thumping heart of perfect-game murders, just waiting to destroy today’s bid for unlikelihood. And the more the probability varies from one inning to the next, the less likely perfect games get, even if we keep the average probabilities the same.

What that means is, without beating you over the head with more numbers, is that it’s perfectly reasonable to expect a perfect game to happen once every few years. And if that is the expectation, it’s possible that you’ll get three in the same season, because three in the same season really isn’t all that different from one every three seasons.

However simple this model may have been, it won’t overwhelm the feeling of recognition when I look at the scorecard and see nothing but zeroes. As someone who sat in the stands and watched a complete oddity of nature throw one, no amount of number crunching could take that away from me.

Question of the Day

If you want to know my favorite culprit for the slight uptick in perfect games over the last century, it’s expansion and the longer schedule. There are both more games and more pitchers, meaning there are concomitantly more opportunities for perfect games. The fact that the last two guys to accomplish the feat were Dallas Braden and Roy Halladay demonstrates just how much it is about randomness, and more games means more opportunities for that randomness to pay off. What is your guess?

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Tommy, I think you might have a problem with your 40% or 38% figure. You are probably counting three one-third of innings as one inning, rather than three innings, to begin with.

As for the chance of a perfect inning, why not do it the even easier way, and take OBP (the real OBP, one where reaching on error is a good thing, etc). Chance of a perfect time at the plate for a pitcher is around 66%, so a perfect 1-2-3 is .66^3 = .287. A perfect 27 is .66^27 is around 1 in 75,000.

Presuming that perfect games occur in slightly more conducive settings, say a true OBP of .300, that makes the above calculation as 1 in 15,000.

With about 350,000 starts in MLB since 1900, that works out to 22 expected perfect starts.
Tom, can you explain what you mean by:

"You are probably counting three one-third of innings as one inning, rather than three innings, to begin with."

I excluded all partial innings from the data I ran for Tommy - a pitcher had to retire all three batters he faced that inning to receive credit for a "perfect" inning pitched.
Colin, I meant for the denominator (opportunity). The way I read Tommy, if someone pitched three 6.1 innings, that would count as 19 innings, when it should count as 21.
I see what you mean, Tom.

It's an interesting question, really, and one that I think comes down to your choice of models. As I hoped to make clear in this piece, it's hard modeling something complex like a perfect game, and a lot of your answer is going to depend on your model. I offered a rather simple model, with plenty of assumptions that aren't particularly rigorous.

So while I agree the denominator I used (the sum of all starters innings) is probably artificially low, I'm not sure round up every partial inning is the best way to do it either. The reason is because many pitchers are taken out earlier if they have previously given up a hit, walk or run in the game than they would be if they had a perfect game intact. That means that it's not always the case that fractional innings aren't completed due to failing to be perfect. Something in between would probably best capture the truth.
Right, the correct answer would have to be between the two. But, I would think it's far closer to me, because I would be shocked if you can find 10% of starting pitchers who were removed mid-inning who did not allow a runner to reach base.

So, the reality is that virtually all mid-inning removals occur when the inning is no longer perfect, and therefore, you have to count that as "1" opportunity, and not "1/3" or "2/3" as your model would specify.
I think you missed something else in the calculation. You have identified the odds of pitching a 1-2-3 inning but you need to work in the odds of pitching two consecutive 1-2-3 innings, the odds of pitching three consecutive 1-2-3 innings, the odds of... etc... to get an accurate measure of probability. Every pitcher manages a 1-2-3 inning occasionally (hopefully more often than that) but following that with another, and another, and another... becomes increasingly difficult.
If I pitched a 1-2-3 inning in the first, how does that affect the probability I throw a 1-2-3 inning in the second?
Agent is absolutely right, as evidenced by followup comments.

In order to have a 38% chance of a 1-2-3 innings, the average OBP of the three hitters is .275. This is going to happen on a combination of good pitchers and bad hitters. In a followup inning, you will still haev the good pitchers, but you will have much better hitters.

So, it would be better if you look at a perfect 9 outs, so that you take away the lineup order as an issue. Indeed, you can start with just who started the game with a perfect 9, count the number of times that happened, and divide that by the number of starts made.

Let's guess this number is going to be 3%. That is, 3% of the time, you start the game with a perfect 9 outs. You raise that number to the power of 3, and you get:

That's about one in 35,000.

And with 350,000 starts since 1900, that gives us the odds at 10 perfect games expected.

Interesting. I took a different tack based upon the observed frequency of perfect games (0.005% of all regular season games from 1901-2010), a figure that works out to one-fifth of Tommy's original estimate, and compared the frequency of those and no-hitters to scoring levels - admittedly not the most granular way to tackle the question - over time:
"A perfect game is a series of nine consecutive perfect innings thrown by the same starter in the same game."

Years ago I had a great seat (about 10 rows up behind the 3rd base dugout) in San Diego for a game between the Padres and Expos. Pedro Martinez pitched 9 perfect innings only to find the game tied at zero. The Expos scored in the top of the 10th but Bip Roberts doubled to lead off the bottom of the 10th and thus I was denied seeing a perfect game hurled by a future hall of famer. Ah well, still a great memory.
I feel compelled, having attended the game, to mention Matt Latos's very-near perfect game last month against the Giants, with his only non-out in 28 plate appearances being an infield single leading off the 6th that was so close that even watching the replay I still couldn't tell if the batter was safe or out. Clearly not in the same category of 'almost' as Galarraga's, but still, we've been shockingly close to 4 perfect games from May 9 to June 3.
"And it’s much more likely that the probabilities vary wildly from inning to inning. Why? Because managers lump their best hitters together in the lineup, meaning that one third of the lineup is the thumping heart of perfect-game murders, just waiting to destroy today’s bid for unlikelihood. And the more the probability varies from one inning to the next, the less likely perfect games get, even if we keep the average probabilities the same."

In our mythical world, the order of the lineup should have no impact on the probability of a perfect game. If you shuffle the best hitters together, it will change the average probability. That is, putting your best hitters together increases the average probability of a perfect inning.
Beautifully well-said.

So if you have 9 guys with the following OBPs:

the chances of getting a perfect three innings is .027. And three groups of those 3 perfect innings is:

Compare that to 27 outs of a .330 OBP hitter:

As you can see, it doesn't really matter how the individual players are spread out.

I have other relevant calculations on my blog for those interested (post 5).
@TangoTiger, in your first comment you compute a rate based on "conducive' conditions. Can't apply that rate to all games since 1900, since all games are not conducive, so the estimate of 22 perfect games, while enticingly accurate, doesn't hold up.

I agree with your latest point, though. If all nine batters are going to bat exactly three times each, and any non-out by anyone zeroes the chance of a perfect game, then the order in which they bat doesn't matter (assuming there's no psychological effect of mounting outs etc). The idea that it does is probably a confusion with the idea that it affects the probability of stringing several events together and thus scoring, but we're not talking about shutouts here.
You are right about your first point, and I made a response yesterday on my blog to that effect.

I also put in more calculations on my blog that are more enticing.
Has a "walk-off" no-hitter ever happened? I.e., scoreless tie through the top of the ninth, with the home team pitcher carrying a no-hitter, and then his team scores to win it in the bottom of the inning, giving him a no-hitter while he's (presumably) sitting in the dugout?
Sort of. The closest I could find was actually a combined 10-inning no hitter (Francisco Cordova pitched the first nine innings and Ricardo Rincon pitched the 10th), and then Mark Smith hit a walk off home run in the bottom of the tenth.

Here's the box score. I bet that would have been a fun one to be at.
Thanks Tommy. Must've been a good celebration after the walk-off homer, unless they were unsure who to celebrate with (thinking of the Robin Ventura walk-off GS, where half the Mets mobbed him at first and half gathered at home plate). I also particularly like that after that game, PIT and HOU were in a virtual tie for first place in the NL Central... under .500, more than halfway through the season.