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.