I think one of the most fun parts of baseball is this part of the year; as we wind down, you can start to root for unlikely things to happen. For example, I’m kind of hoping the Pirates manage to lose at an .800+ clip and keep their sub-.500 streak alive. I’d love to see the Royals make the playoffs. Finally, I’d love to see Yasiel Puig win the NL batting title.
The rules of the game are that you have to have 502 plate appearances to win a batting title. If you’re short, you’re given an 0-fer for the rest. So if Puig finished with 492 PAs, he’d take an 0-for-10 for the purposes of the batting title. Right now, Puig is projected by STEAMER to finish the year with 435 PAs. We’ll accept that number for now, but given that number, let’s think about how likely it is that he has a high enough batting average to win the title.
The first step is to figure out the mark he needs. Let’s go with STEAMER again, and we see Michael Cuddyer, Joey Votto, Yadier Molina, and Chris Johnson all projected to finish at about .320. Let’s assume that one of those four players finishes right at his 87.5% projection (the middle of the highest quartile)…I’ll say Joey Votto, who is projected to go .302 for the rest of the year (the highest of the bunch). Using the binomial distribution, there’s a 16.2% chance Votto finishes 51/149 or better given his “true” .302 batting average. We’ll say that that is the target Puig has to reach: Votto (or one of the others) adds something like 51/149 to his current stats, for a .329 batting average.
What are the chances Puig reaches that clip? To keep it simple, let’s assume STEAMER is right on the number of PAs, ABs, and Puig’s true chance of getting a hit, and then figure out Puig’s chance of getting enough hits to finish at .329 or better. He’s going to end the year with 435 PAs and 390 ABs, if he keeps up his current pace. To that, add an 0-for-67 to get him up to 502 PAs. So he needs enough hits to have a .329 batting average in 457 ABs. That number is 150. He currently has 85 hits in 224 ABs, so for the rest of the year he needs 65 hits in 166 ABs.
Given that STEAMER projects a .293 batting average for the rest of the year, it’s pretty unlikely that he’ll hit at a .392 clip. In fact, his chances of doing so are only about 0.4%, using the binomial model.
What could help his chances? First, there’s no guarantee Votto/Johnson/Molina will get hot enough to make the mark .329. If we drop the required average to .320, using the same method as above, he’d only need 146 hits, which raises his chance to about 2.3%.
Another possibility is that he’s a better hitter than STEAMER projects. If he only regresses to .310, which would make him one of the better hitters in the league admittedly, he has about a 1.6% chance of winning the batting title. And if he is truly a .310 hitter, AND none of the other players near the top of the leaderboard stay hot enough to beat .320, Puig has a whopping 6.6% chance of winning the batting title.
Yeah, I know batting average is stupid. And I know this is a minuscule chance. But isn’t it amazing that Puig has a chance to do something like this at all, after making his debut in June? Baseball!