This year marks the 75th anniversary of two of the most famous feats in baseball, Joe DiMaggio‘s 56 game hit streak and Ted Williams hitting .400. With MLB celebrating the hit streak with a documentary on the anniversary of the start of the streak on May 15th, I thought it would be interesting to look at why DiMaggio, who hit .357, nearly 50 points lower than Williams’s .406, had the better chance to earning a long streak.

The first thing to realize is that batting average is conditional probability. In mathematical notation, BA = p(H|AB), the probability of a hit given an at bat. By Bayes rule we can rewrite this as BA = p(AB|H) * p(H)/p(AB). The first term is one. A hit is always an at bat. So BA = p(H)/p(AB). The probability of a hit would be hits divided by plate appearances (PA), and the probability of an AB would by at bats divided by PA. So BA = (H/PA)/(AB/PA) = H/AB, which is the definition of batting average.

The point is the PA is unit of offense we want to measure against. Batting average is designed to award batting titles. It rewards hits without penalizing walks. In calculating the probability of hits, however, we want to look at p(H), H/PA.

This spreadsheet shows some stats for the two players that speak to the streak. First, notice that despite the lower batting average, DiMaggio picked up more hits than Williams that year. He did it in just a few more plate appearances, so the probability of DiMaggio getting a hit in a given plate appearance was slightly higher than Williams. This naturally works out to higher probabilities of getting at least one hit in a 4 PA game, and and 5 PA game. As these probabilities get multiplied together to calculate the probability of a streak, Williams’s odds come down a lot quick than DiMaggio’s.

Next note that neither player struck out much that year. That allowed them to maximize their skills putting the ball in play hard and take advantage of any poor fielders on the diamond. Williams struck out just 27 times, but DiMaggio was even better with just 13 K.

The number of balls in play (AB-K) finishes the picture. Including home runs, Williams put the ball in play 429 times to 528 for DiMaggio. The Yankee Clipper didn’t walk as much as the Splendid Splinter. Compared to Williams, DiMaggio was a hacker. Williams probably had a number of games where he went 0 for 1 with three walks. That’s a good day on offense, but it’s tough to keep a hitting streak alive when you don’t put the bat on the ball. DiMaggio was constantly putting the ball in play, and enough of them found holes to keep the streak alive for 56 games.

Williams high walk total helped keep his ABs low, so his hits made for a higher BA. DiMaggio’s superb hacking contributed to the streak happening, and in fact Joe was the more likely of the two (if only by a hair) to complete that feat. For long streaks, bet on the high-hit hackers, not necessarily the high batting average.