Behind the Scoreboard July 24, 2009
Perfect Games and Probabilities

As everyone is surely aware, Mark Buehrle pitched baseball's 18th perfect game yesterday afternoon. Now that Buehrle has joined one of baseball's most exclusive clubs, let's see where he fits in. Buehrle is an outstanding pitcher, but not one of the game's all-time greats, and likely not a Hall-of-Famer. However, the group of players to throw a perfect-game ranges from legends (Cy Young and Randy Johnson) to scrubs (Charlie Robertson). Was Buehrle's feat a mere fluke, or did he "deserve" to throw a perfect game.

A very simple analysis shows the probability of throwing a perfect game in one's career. Taking each pitcher's opponent's on-base-percentage and adding the percentage of players reached on errors we can estimate the probability of a hitter reaching base. Using the following formula, we can see the probability of throwing a perfect game as the following:

Probability of Perfect Game = (1-%onbase)^27

And we can use this number and the number of games started to estimate the probability that the pitcher throws a perfect game over his entire career:

Probability of Perfect Game in Career = 1-(1-probPerfect)^#GS

Of course, this assumes that the probability of throwing a perfect game is equal in each of a pitcher's career games, which is not true. A player usually has a peak in which the probability of a perfect game is higher, and thus the formula underestimates the probabilities especially for pitchers who had a peak much higher than the rest of his career, such as Sandy Koufax, Randy Johnson, or Cy Young (actually Young remained quite consistent, but his chances were much higher in the latter half of his career due to the context of the game).

For what it's worth here is a quick list of the 16 modern-era pitchers to have thrown a perfect game, and their rough chances of doing so.

In general, the probability of throwing a perfect game is very low, so all perfect games are "flukes" to some extent. Even a great like Cy Young only had about an 8% chance to throw a perfecto in his career during all of those games he pitched.

As we can see, Mark Buehrle is one of the more unlikely pitchers to have thrown a perfect game. Despite having a very good ERA+, the high scoring era in which pitches makes it difficult to throw a perfect game.

At the top of the list is Addie Joss, but Cy Young should be. He is unfairly hurt by the formula for having pitched in a hitters environment in the first part of his career, raising his career OBP. Taking the second half of his career alone, his probability of throwing a perfect game is over 8%.

There are a few other things that stand out. For being an above average, but not fantastic pitcher, Catfish Hunter enjoyed a very high probability of throwing a perfect game. His Achilles' heel was the homerun ball, which hurts effectiveness as a pitcher but doesn't much affect the chances of throwing a perfect game. He also enjoyed a pitcher's environment.

The luckiest pitcher to throw a perfect game, not surprisingly, was Charlie Robertson who threw a perfect game for the 1922 White Sox. At 49-80 and a 90 ERA-plus, he wasn't great, but he sure had his moment in the sun. Still, at least he had a career - the list of players who have thrown simply a no-hitter is littered with players far inferior to Robertson.

Nevertheless, throwing a perfect game is a rare feat, and anyone who was there yesterday afternoon will have memories to savor for a lifetime.

Other than advances in field maintenance and gloves, can you account for why there have been an increasing number of perfect games the past 45 years (nine in the last three decades and 12 overall) vs. the previous 60 (3)?

Good question Rich. I can't. I think it's particularly surprising that there have been 5 since 1994. It's a hitters era and 5 perfectos in 15 years is a lot. I think luck has a lot to do with it - it might be just random variation. You don't see the same pattern with no-hitters - it's about evenly split between the first and last 55 years. I think that backs up the luck theory.

Ok...now apply this to some current pitchers who are really good? I'd like to see what active pitchers have the highest probability of throwing a perfect game. That would be petty awesome.

I imagine Johan Santana or Tim Lincecum might have the highest probability of doing so.

Curiously...What's the probably of Jonathan Sanchez tossing one?

Jonathon Sanchez's recent no hitter is a very good illustration of the pitcher-independent luck factors that determine perfect games. He pitched to 28 batters, walked or hit none. One batter reached base on an infield error (an error that was an obvious juggle and drop but which also raised the question of whether Uribe would have gloved the ball but gotten a throw to first too late if he had played the natural hop rather than charging). One batter failed to reach base when Aaron Rowand made a wall-crashing catch of a crushed line drive. Buehrle's perfect game was preserved when a home run was plucked away by a leap above the yellow line. I don't know if there were other major non-pitcher factors. But the Uribe error made me reflect on the question how imperfect was Sanchez's performance? Because a *perfect* perfect game would be 81 pitches, all strikes. Or 27 pitches, all made-to-order outs. But a 27 pitch perfect game could, in theory, consist entirely of crushed line drives turned into outs by wall-crashing, above-the-yellow-line grabbing plays. I am not sure the ratings above really address the pitcher's perfection.

Maybe I am missing something, but if I use your formula

Probability of Perfect Game = 1-(1-%onbase)^27

and assume a .333 %onbase then 1 - .333 = .667. Then .667^27 = 0.0000178401. Then 1 - 0.0000178401
is 0.9999821599.

Good catch, the only thing missing was my brain when I typed the formula. As you probably guessed, just drop the "1 -" part.

As for Jonathan Sanchez....at this point, his probability of having thrown one is .00033 - about 4 times less likely than the unlikeliest pitcher to have thrown one. Too bad about that error, but of course, that's factored into the equation.

More people have been to the moon than have pitched a perfect game. A starting pitcher might get 30 starts a year. Among the pitchers to throw a perfect game on pitcher has thrown a perfect game twice. The actual occurrence of a perfect game occurs about one out of 11,000 games. How would you calculate the odds of the occurrence assuming one in 11,000 games is accurate?

Also, even less players have hit 4 home runs in one game with 162 chances every year. What would the probability be for hitting four homers in a games and the odds of the occurrence.

Mark Buerhle will be an interesting case if his career continues injury free for another 12 years or so. He could challenge 300 wins and with a no-hitter and perfect game under his belt, he may only need 250+ wins to get in.