Warning: This post involves math.

In previous analyses of the contact play based on run expectancies, the conclusion was that with a runner on third and one out, having the runner break for home as soon as he sees that the ball will not be caught in the air is a good decision if you expect the runner to be safe at home at least 11% of time that it is fielded by an infielder. With runners on second and third, the success rate can drop to 6% with the contact play still being the better decision. The question then becomes what is the expected probability of a runner being thrown out at home when running on contact, with a ball fielded by an infielder?

While there may be data that could shed some light on this deep in the bowels of Baseball Info Solutions or FIELDf/x, that data isn't freely available to the public. So we'll try the next-best thing - create a model, plug in some assumptions based on what data we can get, and go from there.

If a ball is hit on the ground, either it gets through the infield or it doesn't. If it gets through the infield, it doesn't matter whether the runner goes on contact or not - he'll score. What we care about are the balls hit on the ground that don't go through the infield.

The median fielding percentage of middle infielders is right around .975. While fielding percentage isn't the best statistic to capture true defensive value, that's not what we want to use it for. it does tell us what percentage of balls that a fielder can get to with ordinary effort result in an out.

So we can start building the model by dividing the balls hit on the ground into two groups: one group is those that are reached by the infielder with "ordinary effort", and the other group is those that are reached with what we could call "extraordinary effort".

The official rules of baseball don't give a quantitative guideline for "ordinary effort", so that's our first assumption: Assume that "ordinary effort" is an 80th-percentile, so 80% of balls reached by an infielder are those that can be fielded with "ordinary effort", while 20% are fielded with "extraordinary effort".

For the ground balls fielded with ordinary effort, we can use the median fielding percentage to assume what percentage result in the batter being thrown out at first if the throw goes there: 97.5%.

For the ground balls fielded with "extraordinary effort", let's assume 50/50 - 50% chance of throwing the batter out at first, 50% of the batter being safe.

But that doesn't actually tell us the probability throwing a runner out at the plate; for that, we need to do a little more model-building.

Throwing a runner out at first can be modeled as requiring three things: the fielder fielding the ball; the fielder making an accurate throw to first; and the first baseman catching the throw. Throwing a runner out at the plate requires four thing: the fielder fielding the ball; the fielder making an accurate throw to the plate; the catcher catching the throw; and the catcher making the tag. In both cases, there is also a time factor, in that these things have to take place before the runner safely reaches the base.

In other words, to throw a batter out at first base, P(field) * P(throw-1B) * P(catch-1B) * P(timely-1B) = 0.975.

The probability of throwing out a runner at home can be written as P(field) * P(throw-H) * P(catch-H) * P(tag) * P(timely-H), and each of these factors (except P(tag)) calculated relative to the equivalent factors for throwing out a runner at first base. We don't have to know the actual probability of a fielder making a given play; we instead assume the relative probabilities of making the corresponding plays to first base and to home.

For example: Regardless of whether the fielder is going to throw home or to first, the probability of him fielding the batted ball can be assumed to be the same. (There may be reason to believe a fielder is less likely to cleanly field a batted ball when they see the runner breaking for the plate because they are distracted or rush the play, but we'll disregard that.)

The probability of making an accurate throw depends on the distance of the throw and the target area. The distance of the throw depends on the fielder making it: A pitcher is equidistant from first base and home plate. A shortstop playing in is about 100 feet from either home plate or first base. A third baseman playing in is about 90 feet from home plate but about 110 feet from first base; a second baseman playing in is about 100 feet from home plate but only about 60 feet from first base; while a first baseman makes no throw at all. Disregarding balls hit to first base for now*, that leaves two situations where the throw is longer and two where it's shorter, so we'll disregard the effect on throw distance for this relative probability.

The target area for a throw to first can be viewed as anywhere the first baseman can reach while keeping a foot on the base. (Disregarding special cases of the first baseman coming off the bag to make a catch and either finding the bag again or tagging the batter.) Figure a typical first baseman can stretch 8 feet to his glove side, 7 feet to his throwing side, and 9 feet vertically. To simplify, deform that a bit to a semi-circle of radius 8 feet centered on first base, and that gives you a target area of about 100 square feet.

The target area for a throw to home plate stretches from about ground level to about shoulder high; from about a foot to the first base side of the plate to about two feet to the third base side of the plate. Anywhere outside that area and it becomes too difficult for the catcher to make the catch and apply a tag. That gives you a target area of about 22.5 square feet, or about one-quarter the size of the target area for first base.

However, these are professional athletes who throw baseballs for a living; reducing the target area by a factor of 4 doesn't reduce their probability of hitting it by a factor of 4. We can estimate that such a reduction in target size translates to 75%-80% the likelihood of the fielder hitting the target. Using 80%, this becomes P(throw-H) = 0.8 * P(throw-1B)

We can assume that the first baseman cleanly catches 100% of balls thrown into the target area (disregarding the <1% of 1B fielding errors). That's his job. However, a catcher isn't going to cleanly catch 100% of throws from infielders with a baserunner bearing down on him. Let's assume he catches 95%, so P(catch-H) = 0.95 * P(catch-1B).

The next factor is the probability that the catcher can apply a tag given he catches the throw. This covers all the messy factors such as whether the baserunner can avoid the tag, barrels over the catcher and knocks the ball out, whether the tag hits the body before the foot hits the base, and so on. Let's be nice to our catcher and assume he successfully gets a tag down 95% of the time when he catches the ball. Because no tag is needed at first, we simply apply this factor (P(tag)) to the calculation.

Finally, there's the question of whether this process actually beats the runner to the base. The runner on third is some distance from the base and taking a walking lead; the batter may actually be more than 90' from first base (if RH) or facing the wrong way (if LH), so the runner will be faster to the plate than the batter to first. An average home-to-first time for a major leaguer is 4.3 seconds (if RH; 4.2 if LH); a fast time is 4.2 seconds (4.1 if LH). There's no data on third-to-home times, but a good first-to-second time for a base stealer is 3.2 seconds. A runner on first doesn't have a walking lead, so a third-to-home time could be faster by a couple tenths of a second, but let's disregard that. The simplest way** to use this data to get a relative probability of getting the ball home in time to tag the runner compared to getting the ball to first in time to beat the batter is to just take the ratio of the two: 3.2/4.2, or P(timely-H) = 0.76 * P(timely-1B).

If you plug these assumptions back into the equations above, you get:

P(out at home) = 0.8 * 0.95 * 0.95 * 0.76 * P(out at first).

We know that a ball fielded with "ordinary effort" results in a 97.5% probability of the batter being out at first; the same ball results in a 53.6% probability of the runner being out at home.

If a ball that is fielded with "extraordinary effort" results in a 50% probability of the batter being out at first, the same ball results in a 27.5% probability of the runner being out at home.

And if 80% of balls fielded by infielders are fielded with "ordinary effort" and 20% are fielded with "extraordinary effort", the total net is 88.7% of balls fielded by infielders on which the throw goes to first will result in the batter being thrown out, while 48.4% of balls fielded by infielders on which the throw goes home will result in the runner being thrown out at the plate.

So now we can apply these probabilities to our favorite Run Expectancy matrices. This time, we'll add the probability (since we've calculated it above) and run expectancies of the batter being safe at first if the runner on third doesn't go.

First, the runner on third with one out.

If the runner goes on contact, there are two possible outcomes: He is out at home (P=0.484), leaving a runner on first with two out (RE = 0.2213), or he is safe at home (P=0.516), leaving a runner on first with one out and a run in (RE = 1.5115). Total run expectancy is 0.8869 runs.

If the runner doesn't go on contact, there are two possible outcomes: The batter is out at first (P=0.887), leaving a runner on third with two out (RE = 0.3634), or he is safe at first (P=0.113), leaving runners on first and third with one out (RE = 1.1462). Total run expectancy is 0.4516 runs.

The contact play gives 0.4353 more expected runs.

Next, runners on second and third with one out.

If the runners go on contact, there are two possible outcomes: The runner is out at home (P=0.484), leaving runners on first and third with two out (RE = 0.4839); or he is safe at home (P=0.516), leaving runners on first and third with one out and a run in(RE = 2.1462). Total run expectancy is 1.3415 runs.

If the runners don't go on contact, there are two possible outcomes: The batter is out at first (P=0.887), leaving runners on second and third with two out (RE = 0.5813), or he is safe at first (P=0.113), leaving the bases loaded with one out (RE = 1.5383). Total run expectancy is 0.6892 runs.

In this case, the contact play gives 0.6523 more expected runs.

In both cases, the contact play basically **doubles **the run expectancy of the inning. There is almost no combination of assumptions that makes the decision swing in favor of not sending the runner on contact. The factor with the largest impact is the "timeliness" factor - but even if you assume the runner on third is so slow that a throw to the plate is 25% *more *likely to beat the runner home than beating the batter to first, the run expectancy is still higher for the contact play.

I've got all these assumptions built into a spreadsheet, so if you'd like to see the results with different assumptions, put them in the comments and I can re-run the numbers.

* Because we're using the process of field/throw/catch to estimate the relative probability of throwing out a runner at the plate, and that process doesn't apply on balls hit to first base.

** Not necessarily the most mathematically rigorous, but it's getting late.

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