How Valuable Is a Double?

(...Or a walk, hit-by-pitch, single, triple, home run, stolen base, sacrifice, or non-strikeout?)

As I claimed yesterday, the value of a plate appearance is assessed by the number of bases a hitter moves himself and his teammates. Well, it's pretty easy to tell the number of bases a batter moves himself with any one of those actions. The hard part is figuring out how far teammates advance in an average plate appearance through each of those actions.

While watching a game, it's pretty easy to tell. When Mark Teixeira doubles home Derek Jeter from 3rd and Johnny Damon from 2nd, he's created 5 Bases Advanced (2 for himself, 1 for Jeter, and 2 for Damon). But we want to know how much a generic double is worth. In order to do that, we need to invoke some basic baseball knowledge and a little math.

The basic idea is this: there are 8 possible baserunning arrangements for a batter when he comes to the plate (nobody on, man on 1st, man on 2nd, man on 3rd, men on 1st and 2nd, 1st and 3rd, 2nd and 3rd, and the bases loaded), and we can predict how many total bases all runners will advance from each arrangement for a particular outcome at the plate. What we need to do is determine the probability that a hitter will come up in each of those situations, and then find the average number of Bases Advanced.

Take a double, for example. Obviously, the batter advances himself 2 bases. And the runners?
Nobody on: Uhh, there's nobody on. 0 Bases.
Man on 1st: Advances to at least 3rd base, scores 50% of the time. 2.5 Bases.
Man on 2nd: Scores. 2 Bases.
Man on 3rd: Scores. 1 Base.
1st and 2nd: Guy on 2nd scores, guy on first scores half the time. 4.5 Bases.
1st and 3rd: Guy on 3rd scores, guy on first scores half the time. 3.5 Bases.
2nd and 3rd: They both score. 3 bases.
Loaded: Guys on 2nd and 3rd score, guy on first scores half the time. 5.5 Bases.

But wait. We're not done. Now we need to know the odds that a batter comes up with each of those arrangements.