smastroyin said:
The proper way to think about it is to think about all of the scenarios.
.643 is the chance with first and second and none out.
Now you have a batter. Several things can happen as a result of this at bat. If we are talking only about the need to score at least 1 run:
Batter reaches safely while a runner scores. Obviously this increases the chances to 1.000
Batter reaches safely while all runners advance exactly one base. This increases the chances to .877 (bases loaded, no out)
Batter makes a productive out that advances both runners. This is the .698
Batter makes a productive out that advances one runner. This is .652 (runners on the corners with one out)
Batter makes a non-productive out. This reduces the chances to .429 (runners on first and second, one out)
Batter hits into DP that takes middle runner. This reduces to .270 (runner on third, two out)
Batter hits into DP that takes lead runner. This reduces to .230 (runner on 2nd, two out)
Batter hits into DP that takes both runners. This reduces to .135 (runner on 1st, two out)
Batter hits into TP that ends inning. This reduces to 0.
I don't feel like doing all the math, so let's presume there are only three swing-away scenarios and that the bunt is successful.
Batter reaches safely and a runner scores. Let's say the chance of this is .200
Batter reaches and no runner scores. Let's say the chance of this is .200
Batter makes a non-productive out. So the chance of this is .600
.2 * (1) + .2 * (.877) + .6 * (.429) = .6328
So the expected outcome of letting the man swing away is a 63.3% chance of scoring at least a single run.
The bunt puts you at a 69.8% chance, so purely by the numbers in this case, you bunt.
However, if you only successfully bunt at a 70% rate, the bunt is really only worth
.7 (.698) + .3 (.429) = 61.7%
I'm glad that I scrolled down far enough, because I was about to do a lot of this. I think that we can change the expected value a little though, and tweak these probabilities a bit.
I think that one important thing would be including the fourth outcome, batter makes a productive out (which is the same as the intended outcome of the bunt) as one of the possible outcomes.
Looking at the last two years of data from Fangraphs, it looks like we get:
35% chance of non-productive out (K, IFF, etc.)
33% chance of productive out
32% no out, which is split into: 20% no out, no score, and 12% no out score (this is looking at hit type distribution (frequency of doubles, triples, HR, etc. with the man on second scoring on everything but a single automatically, and scoring on a single at the average rate which is almost 60%)
So it becomes:
.12 * (1) + .2 * (.877) + .35 * (.429) + .33 * (.698) = 67.5%
So now, that means that unless you think you can get the bunt executed properly 90% of the time or more (which gives you the 67% chance), you are better off swinging away.
AND, that's assuming that you only need one run (so it's the bottom of the 9th and you are at home), because swinging away has a lot more possible outcomes where you can score multiple runs than the bunt does. So basically, unless you have a perfect bunter, swing away.
Note: Using league averages for hit type distribution doesn't take into account a guy who is a pure singles hitter, which is primarily when you would bunt, but Drew has some good pop, and slugs at least league average.