# Baseball, batting average, and probability

A baseball player's batting average is equivalent to the probability he will get a hit for any given at-bat (at-bats don't include Errors, Walks, or HBP and a few other exceptions). So for a specific player with a specific batting average, the probability that he will get a hit against an unknown pitcher is exactly equivalent to his batting average.

Similar to the AVG statistic for hitters, pitchers have a statistic called Batting Average Against (BAA). This statistic is calculated in the exact same way as hitters except it's done for a pitcher. It's equivalent to Hits divided by At-Bats of opposing batmen (na na na na). So for a specific pitcher with a specific BAA, the probability that he will allow a hit against an unknown batter is exactly equivalent to his batting average against.

Intuitively, it seems obvious to me that a batter, no matter his personal batting average, is more likely to get a hit against a pitcher with a high BAA, and less likely to get a hit against a pitcher with a low BAA. Additionally, a pitcher, no matter his own BAA, is more likely to allow a hit when facing a batter with a high AVG than when facing a batter with a low AVG.

So the question is, given a specific batter with a specific AVG and a specific pitcher with a specific BAA, how do we calculate the probability that that specific batter will earn a Hit against that specific pitcher?

EDIT: It's fair to assume we're talking about MLB, and we have an overwhelming wealth of extra information. Assume we're talking about a batter with thousands of at-bats recorded, a pitcher with thousands of batters faced, and we know all the information about the average league AVG, average league BAA, etc., but this specific batter and this specific pitcher have never faced each other. How would we calculate the probability of a Hit?

EDIT2: Let's not get bogged down with vsLHP, vsRHP, RISP, and other statistics. These are merely statistics that can be used to give a more accurate probability. The method for calculating the probability should remain basically the same. Let's just suppose we have Batter A who has an AVG of .300, and the average BAA of the pitchers he's faced (weighted to account for facing some pitchers more frequently etc) is .250. And we have Pitcher B who has a BAA of .225, and the average AVG of the batters he's faced (again, weighted) is .250. Batter A has never faced Pitcher B before, but both have thousands of At-Bats/Batters-Faced. How do we calculate the probability of a Hit versus an Out?

• BAA is usually called "Opponents' Batting Average" or OBA, in my experience watching MLB. – KeithS Oct 3 '13 at 18:38
• I've seen it referred to both ways. – nhgrif Oct 3 '13 at 18:38
• Sabermetrics folks have already looked into this: see for example here, here, here. – Michael Lugo Oct 4 '13 at 16:32
• @MichaelLugo Ah, thank you. This is what I was looking for! – nhgrif Oct 4 '13 at 16:46

Basically you can't, unless you know a lot more information. For example, suppose we have three batters $a,b,c$ with averages $0.75,0.5,0.25$ and three pitchers $A,B,C$ with BAAs $0.75, 0.5, 0.25$. If we multiply by $600$ at bats by each, $200$ for each pitcher/batter combination you are filling in a matrix $$\begin {array} {c|c|c|c|}&A&B&C&Total\\ \hline \\a&&&&450\\ \hline \\b&&&&300\\ \hline \\c&&&&150\\ \hline \\&450&300&150 \end{array}$$ You have six equations in nine unknowns, so lots of freedom. Each cell can contain any number from $0$ through $200$

Added: compare these two sets of hits: in each case, each pitcher pitches to each batter 200 times. The batting averages and BAAs are the same.

$$\begin {array} {c|c|c|c|}&A&B&C&Total\\ \hline \\a&200&150&100&450\\ \hline \\b&150&100&50&300\\ \hline \\c&100&50&0&150\\ \hline \\&450&300&150 \end{array}$$

$$\begin {array} {c|c|c|c|}&A&B&C&Total\\ \hline \\a&200&200&50&450\\ \hline \\b&200&100&0&300\\ \hline \\c&50&0&100&150\\ \hline \\&450&300&150 \end{array}$$

The first approximates the intuitive view you seem to have. The second supports the same data, and the worst batter is hitting $0.500$ against the best pitcher.

• I don't fully understand this (sorry, I'm not a math person... I posted the question because I want to understand the math more because I'm interested in developing a simulation program regarding baseball). But in my question, we can assume that we're talking about MLB. We have a specific AVG for the batter and a specific BAA for the pitcher. We also can find the average AVG for the league (and the average BAA, but this should match average AVG, I should think, right?). So is there a way to tell by comparing the batter, pitcher, and the league averages? – nhgrif Oct 3 '13 at 17:22
• For example, if the batter's AVG is .300, but the league average AVG is .250. And the pitcher's BAA is .225, but the league average BAA is .250. We can also assume we know the total number of batters, pitchers, or even simply total number of At-Bats for the league, as well as the specific number of At-Bats the batter has had and the specific number of Batters Faced for the pitcher. – nhgrif Oct 3 '13 at 17:23
• The point is that I can put numbers in the table in many ways to make the row and column sums come out right, which will give all the players the correct statistic. You seem to be assuming that a high average batter will do better against all pitchers than a low average batter, but that may well not be true. Maybe the high average batter hits right handed pitching very well, the low average batter hits lefties well, and there are lots more right handed pitchers. – Ross Millikan Oct 3 '13 at 17:29
• I know there are other factors. But for argument's sake, if Batter A has higher numbers in every single batting statistic regardless of any factor (vsLHP, vsRHP, RISP, etc) then Batter B does, it seems only natural that he will be more likely to earn a Hit versus any random pitcher. Because Batter A is a better player than Batter B. For example... Derek Jeter is more probably to earn a hit in an at-bat against Cliff Lee than, well... me. Because Derek Jeter is a better baseball player than I am, and if I were a MLB player, the statistics would show that. – nhgrif Oct 3 '13 at 18:23
• So how would I go about quantifying the probability of Derek Jeter scoring a hit against Cliff Lee versus the probability of myself scoring a hit against Cliff Lee? The fact of the matter is, Derek Jeter has probably see at-bats versus Cliff Lee, so there's probably already a Derek Jeter vs Cliff Lee AVG/BAA. But assuming they had never faced each other, and Derek Jeter has thousands of at-bats, and Cliff Lee has thousands of batters faced. We know everything there is to know about these two players as well as the league as a whole. How do we quantify the probability of Jeter earning an H? – nhgrif Oct 3 '13 at 18:25