Deep in the weeds (

http://www.mfooz.com/bblog/wp-content/u ... efense.pdf):

A. How Many X-Rolls?

Our question in this section is: In the time it takes a player to get 108 PA rolled off his hitting card, how many X-rolls on defense can we expect him to get?

Well, he only gets his card rolled half the time that he's up; that gets us up to 216 PA. And since he has eight teammates and they all bat in order, he only hits 1/9 (11.1%) of the time. That gets us up to (216 / 11.1%) = 1,946 total rolls, in order to hit that one batter's hitting card 108 times.

Since all SOM pitchers have the same exact X-rolls, we know precisely what the chance of each X-roll is, just by knowing dice probabilities. (These are provided by Madsen. They are the total probabilities, not the probabilities given that the roll is off the pitcher's card, so we don't need to halve that 1,946.)

Table 1: Probabilities of Each X-Roll

1B 2B 3B SS LF/RF CF P

0.93% 2.78% 1.39% 3.24% 0.93% 1.39% 0.93%

Multiply all of those by 1,946, and you get:

Table 2: Number of X-Rolls Per 108 Offensive PA

1B 2B 3B SS LF/RF CF P

18 54 27 63 18 27 18

That answers our question. We can expect a SS to field 63 (ss) X rolls in the time it takes him to get 108 rolls off his card on offense; a 1B to field 18 (1b) X rolls; etc. In the defensive NERP allowed formula -- which again is

(TB * .318) [- (.333 * DP)] + (OB * .25) - (AB * .085) -- Table 2's numbers will be our “AB” for each position.