# Never Make the First or Last Out at Third Base … Perhaps

This article was written by Ryan Gantner

This article was published in Spring 2016 Baseball Research Journal

Baseball players, even those playing as children, have likely heard the familiar adage Never make the first or last out at third base. This advice warns players to exercise extreme caution when deciding to advance to third base when there are presently zero or two outs, imploring them to remain at second unless successful advancement is virtually certain. But are there data to support this wisdom? In this paper, we explore the soundness of the advice in this adage, including various interpretations of it, by looking at Major League Baseball data in various ways.

EXPECTED NUMBER OF RUNS GENERATED PER INNING

One way to determine whether advancing to third base is prudent is to examine the expected number of runs scored in each of three possible scenarios: the runner successfully advances to third base, the runner is out while attempting to advance to third base, and the scenario in which the runner does not attempt to advance and remains safely at second base. One can examine the average number of runs scored per half inning in each of these situations by looking at MLB data from an entire season. These data are readily available. For instance, the Baseball Prospectus website contains the data in Table 1 based on the 2014 MLB season.1

Table 1

Baserunners Exp. runs,
0 outs
Exp. runs,
1 out
Exp. runs,
2 outs
0 0.4552 0.2394 0.0862
3 1.2866 0.8873 0.3312
20 1.0393 0.6235 0.2901
23 1.8707 1.2714 0.5351
100 0.8182 0.4782 0.1946
103 1.6496 1.1261 0.4396
120 1.4023 0.8623 0.3985
123 2.2337 1.5102 0.6435

The expected number of runs in a half inning from various base runner configurations. Data are from the Baseball Prospectus website. The presence of a runner on a base is indicated by that base number appearing in the configuration code; there is a 0 in that spot if there is no runner at the base.

Using tables like Table 1 we can compute threshold values for the probability of successful advancement to third: the probability of successful advancement which would yield identical expected values for runs scored for both situations (the runner attempts to advance and the runner makes no such attempt). Anytime the actual probability of success exceeds the threshold value, the expected number of runs scored will be maximized by having the runner attempt to advance to third base. These probability calculations are done assuming that the future events (such as the number of runs scored in subsequent play) in the inning are independent of past events, given the current baserunner configuration. In reality, future events are dependent on who is batting, who is running, who is pitching, where the game is played, the weather, etc. Modeling probabilities based on these dependencies is not only beyond the scope of this paper but also not in the spirit of a simple adage that advises one to never make an out at third. Furthermore, work done by David W. Smith suggests that considering these dependencies actually does not drastically change the transition probabilities anyway.2

There are two situations to examine. One situation is when a runner is attempting to advance to third base and this runner is the only baserunner. This situation could arise when a batter attempts to stretch a double into a triple, when a runner on second tries to steal third base, or when a runner on second attempts to advance to third on a fly ball or when a pitch gets past the catcher. When there is nobody out, the expected number of runs per half inning is 1.2866 when there is a runner on third and nobody out and 0.2394 when there are no baserunners and one out (which would occur if the runner were unsuccessful in advancing to third). Thus, if the probability of successful advance to third base is denoted by p, the expected number of runs by attempting to advance to third base with nobody out is 1.2866p + 0.2394(1-p). Setting this equal to 1.0393 (the expected number of runs scored with a runner on second with nobody out, the situation in which no advance is attempted) and solving for p gives the threshold value of p for this situation; in this case p=0.764. A runner will increase the team’s expected number of runs scored in the inning by attempting to advance from second to third if he anticipates a probability of success better than 0.764. Other threshold values appear in Table 2.

Table 2

2014 thresholds 0 outs 1 out 2 outs
one base runner 0.764 0.671 0.876
also runner on 1st 0.789 0.717 0.907

Threshold values for advancing from second to third base, based on 2014 MLB data.

As we can see by Table 2, the threshold values for going to third base are higher with zero or two outs than with one out. This does give some evidence to support the adage. This is true whether or not we assume that there is a runner behind the one who is attempting to advance to third (which we assume will stay at first base regardless of whether the other runner attempts to advance to third).

To show that these thresholds are not a fluke, a similar analysis can be done using the 2013 MLB season. The threshold values obtained appear in Table 3:

Table 3

2013 thresholds 0 outs 1 out 2 outs
one base runner 0.774 0.680 0.866
also runner on 1st 0.800 0.724 0.899

Threshold values for advancing from second to third base, based on 2013 MLB data.

While this does support the adage, it is worth noting that the thresholds with zero or one out are not extremely different.

It is worth mentioning that this particular analysis was done with older MLB data (not immediately clear which year or whether it was an entire season or the whole league, etc.) in a series of two blog posts by Zachary Levine of the Houston Chronicle in 2009.3, 4

In those articles it was mentioned that the thresholds for going to third with zero outs and with one out are fairly close to each other, while the threshold for going to third with two outs is substantially greater. With the 2013 and 2014 data, that effect is not quite as pronounced. Levine even offers a change to the axiom to say:

Don’t make the last out at third (The first is forgivable).

It is also interesting to note that numerous analyses involving the expected number of runs after various baserunning transitions are done in the work of David W. Smith. While Smith doesn’t specifically address this particular threshold calculation, he does specifically mention it as one that could be performed using this type of analysis.

EXPECTED NUMBER OF RUNS FORFEITED PER INNING

Also mentioned in the Houston Chronicle blogs is that rather than examining the expected number of runs generated per inning by each of the three baserunner configurations that could arise, we could look at how many runs per inning we expect to forfeit if the runner attempts to advance to third base but is unsuccessful. For instance, in 2014 the mean number of runs per half inning scored after we reach a situation in which there is a runner on second and nobody out (the runner does not try to advance) is 1.0393 and the mean number for nobody on and one out is 0.2394. Therefore, a runner being thrown out at third for the first out has squandered 0.7999 runs. Table 4 shows the mean number of runs forfeited with each attempt at advancing to third.

Table 4

0 outs 1 out 2 outs
successfully advance to third, lone runner 1.2866 0.8873 0.3312
unsuccessfully advance to third, lone runner 0.2394 0.0862 0
no attempt to advance to third, lone runner 1.0393 0.6235 0.2901
runs forfeited by unsuccessful advance 0.7999 0.5373 0.2901
successfully advance to third, also runner on 1st 1.6496 1.1261 0.4396
unsuccessfully advance to third, also runner on 1st 0.4782 0.1946 0
no attempt to advance to third, also runner on 1st 1.4023 0.8623 0.3985
runs forfeited by unsuccessful advance 0.9241 0.6677 0.3985

The expected number of runs in each situation and the number of runs forfeited if the advancing runner is unsuccessful.

In both situations, while the threshold for advancing to third with two outs is quite high (we need to be all-but-certain of a successful advance in order for our average number of runs to increase), the number of runs lost by an unsuccessful advance is actually substantially less with two outs than with one or no outs. This, together with our previous skepticism about the wisdom of not making the last out at third base, might seem to indicate that we should say:

Never make the first out at third base.

MAYBE WE ONLY NEED ONE RUN

The previous two subsections contain arguments based on the increase or decrease in the expected number of runs scored from various situations. However, there are many times in baseball that we don’t need to maximize the expected number of runs scored in the inning, rather we would like to maximize the probability of scoring at least one run (or at least 2 runs, etc.). To perform this analysis, the author wrote a computer program to scour through the play-by-play files provided by Retrosheet.org to determine the fraction of the time from each situation when at least one run is scored.5 The results are presented in Table 5.

Table 5

situation at least
1 run
at least
2 runs
at least
3 runs
at least
4 runs
000, 0 out 0.255 0.114 0.049 0.020
100, 0 out 0.404 0.229 0.108 0.046
020, 0 out 0.609 0.259 0.112 0.050
003, 0 out 0.857 0.283 0.125 0.047
120, 0 out 0.596 0.379 0.222 0.109
023, 0 out 0.849 0.573 0.243 0.129
103, 0 out 0.876 0.426 0.251 0.112
000, 1 out 0.147 0.057 0.022 0.008
100, 1 out 0.258 0.140 0.056 0.020
020, 1 out 0.384 0.150 0.059 0.023
003, 1 out 0.660 0.166 0.071 0.028
120, 1 out 0.398 0.234 0.133 0.049
023, 1 out 0.692 0.390 0.163 0.070
103, 1 out 0.620 0.255 0.139 0.055
000, 2 out 0.061 0.018 0.006 0.002
100, 2 out 0.122 0.040 0.013 0.004
020, 2 out 0.206 0.065 0.025 0.009
003, 2 out 0.255 0.059 0.021 0.006
120, 2 out 0.210 0.106 0.058 0.016
023, 2 out 0.234 0.185 0.062 0.019
103, 2 out 0.249 0.102 0.049 0.014

The fraction of the time each event occurs subsequently in the same inning when the baserunners are in a particular situation. For instance, when a runner is on third base (and only third base) with one out, at least one run is scored 66% of the time.

We can then perform a similar analysis to what we did before. For example, suppose we would like to maximize the probability of scoring at least one run. With one out, a batter gets a good hit and is trying to decide whether he should remain happy with a standup double, or try to extend to a triple. If he is successful in his advance to third, the probability of scoring at least one run is 0.660. If he is unsuccessful, the probability drops to 0.061. If he doesn’t attempt to advance, the probability is 0.384. Again assuming that what occurs after this batter is independent of how the batter runs the basepath, the runner should try to advance to third if his probability of success is at least 0.54. We can do a similar analysis of each situation, which yields the probabilities in Table 6:

Table 6

situation \ runs needed 1 run 2 runs 3 runs 4 runs
go to third, 0 outs, lone runner 0.651 0.896 0.874 1.063
go to third, 1 out, lone runner 0.540 0.893 0.813 0.798
go to third, 2 outs, lone runner 0.806 1.103 1.216 1.359

go to third, 0 outs, runner on 1st 0.547 0.834 0.850 0.968
go to third, 1 out, runner on 1st 0.553 0.901 0.952 0.874
go to third, 2 outs, runner on 1st 0.843 1.036 1.180 1.128

The critical probabilities for advancing to third if we need a certain number of runs. If the probability the runner will successfully advance to third is greater than these values, advancing to third will improve the probability of achieving the desired result. Values are based on 2014 MLB data.

The most striking feature of the values in Table 6 is that some of the probabilities are more than one. For example, sending a runner to third base with two outs actually decreases the team’s probability of acquiring two or more runs in the inning, based on 2014 data. The only logical explanation that the author can think of for this is that with a runner on second, the opposing team may be tempted to walk the next batter to yield a force play, and this strategy may backfire. According to this analysis, runners should stop at second base with two outs in this situation, even if a triple were guaranteed!

In Table 6, we notice that with a lone runner the critical probabilities for advancing to third are lower with one out than with two or zero, regardless of how many runs are needed in the inning. This may speak to the adage in the sense that attempting to advance to third base is less risky with one out than with zero or two. However, in most cases there is not much difference between the critical probabilities for zero outs and for one out. Furthermore, with a runner on first the critical probabilities are actually higher (except if the team needs four runs) with one out than with zero outs. Perhaps most importantly, the only values in the table which are less than 0.8 are when attempting to advance to third with zero or one out and only one run is needed. Perhaps the axiom should say:

When only one run is needed, don’t make the last out at third base. When more than one run is needed, don’t get out at third base no matter how many outs there are.

IS THIRD BASE UNIQUE?

What’s so special about third base? Perhaps the axiom should read Don’t make the first out at second base, or Don’t make the third out … anywhere! In this subsection, we’ll examine whether there is anything different about third base that makes it worthy of such advice

Using a similar analysis, we compute the threshold values for going to second (stretching a single into a double, stealing second, advancing from first to second on a wild pitch, etc.). We also compute the threshold values for attempting to score from third base (sorted according to where any other baserunners are). The results are displayed in Table 7. For example, in order to increase a team’s expected number of runs scored in the inning, a runner should only advance to second with one out if the probability of successful advance is greater than 0.7296.

Table 7

0 outs 1 out 2 outs
going to 2nd 0.72347 0.72957 0.6708
going home (lone runner) 0.86133 0.69468 0.30492
going home (runner on 1st) 0.87424 0.72569 0.36799
going home (runner on 2nd) 0.88092 0.73594 0.41477
going home (runner on 1st & 2nd) 0.89052 0.75946 0.46014

Threshold values for advancing to selected bases. Computed using 2014 MLB data.

What we see is that the threshold values for advancing to second do not exhibit substantial differences with respect to the number of outs, though advancing to second with two outs has a slightly lower threshold than with 0 or 1. However, the threshold values for advancing to home do vary dramatically: much more dramatically than the thresholds for advancing to third. Because the threshold for advancing to home plate with 0 outs is so much more than the thresholds with other numbers of outs, perhaps our adage should say:

Never make the first out at home plate.

We can also repeat our analysis of the situation where we need a certain number of runs. We calculate the critical values for situations in which we need only one, two, three, or four runs and present them in Table 8. Threshold values for advancing to home are presented in the case when there is only one baserunner; the other baserunner configurations give similar conclusions.

The critical values for scoring at least n runs in an inning are computed as

critical value = p-q/s–q

where

p=P (scoring n runs if no attempt to advance is made)

q=P (scoring n runs if attempt fails)

s=P (scoring n–1 runs if attempt succeeds)

Table 8

situation \
runs needed
1 run 2 runs 3 runs 4 runs
go to 2nd, 0 outs 0.5576 0.8514 0.9509 0.9203
go to 2nd, 1 out 0.6096 0.9239 0.9507 0.8703
go to 2nd, 2 outs 0.5937 0.6171 0.5288 0.4891

go home, 0 outs (lone runner) 0.8319 1.1398 1.1188 0.9430
go home, 1 out (lone runner) 0.6383 1.1474 1.2822 1.2951
go home, 2 outs (lone runner) 0.2554 0.9666 1.1546 1.0564

Critical probabilities for advancing to second base and home plate when a specified number of runs is needed. The data were calculated using 2014 MLB data.

We notice a few things immediately. For example, when only one run is needed the critical value of the probability of successful advance to second base is essentially the same regardless of the number of outs. However, when more than one run is needed, the critical probability for going to second is very high with zero or one outs and much lower with two outs, suggesting that it is rarely worth it to advance to second base in zero or one out situations when more than one run is needed. Also, the critical probability for attempting to score is only reasonably small when one run is needed, and even then it is still high when there is nobody out.

Indeed, many of the critical probabilities in Table 8 are greater than one, indicating that a team is actually more likely to score, say, at least three runs in an inning when the baserunner is kept at third instead of scoring, even if successful advance is certain. The author acknowledges that this advice seems a bit suspicious and offers some interpretation. First, the independence assumption (that the events of the game following a baserunner advance are independent of the past events) is not entirely sound. The presence of a runner on third base may indicate something about the pitcher’s current state, for instance, and may improve the probability of scoring more subsequent runs in an inning, thus driving the threshold probabilities below one. Second, the numbers came from MLB data, so they are based on what MLB teams actually did, not based on the results of some unbiased experiment (there were no “clinical trials” in the way that a medical treatment would get approved). Finally, as the events in question get more specific (e.g., a team scores four or more runs in an inning with a runner on third and two outs), the number of occurrences begins to lessen somewhat, so a smaller sample size may begin to distort the results. In any event, the upshot is that the critical values for advancing to home when more than one run is needed in an inning are extremely high.

We could pen a more elaborate adage:

If more than one run is needed in an inning, never get out at home and never make the first or second out at second base. If only one run is needed, never make the first out at home plate, but making the third out at home is entirely forgivable.

However, the first sentence of our new adage has forgotten the critical values for third base. In fact, if more than one run is needed, the only situation in which it makes sense to advance a baserunner is when attempting to advance to second with two outs. Putting this together, we could say the following:

If more than one run is needed in an inning, the only time a runner should be advanced when success is not completely certain is when going to second base with two outs.

RESULTS BASED ON A MARKOV MODEL

Another way to analyze the situation is to build a model to examine baserunning situations. As is mentioned by Smith in his paper, one way this has been done is to model the game of baseball as a Markov chain.6 We use for states of the Markov chain the combination of which bases contain runners and the number of outs. For example, one state is “nobody on base with one out” and another is “runners on first and third with two outs.” Modeling the game as a Markov chain is natural for our purpose for two reasons. First, as mentioned earlier, the calculation of threshold values presumes the independence of future events from past events, given the present situation. Therefore, we are in effect employing a Markov assumption even in that case. Furthermore, by using a Markov model we are extending the foundation of empirical probabilities to a more generalized context, while still keeping the foundation in realized data.

To determine transitions between these states, we examined play-by-play data from the 2013 and 2014 Major League Baseball seasons. Every single play was examined and tallied in order to assess the probabilities of these transitions during these seasons. For example, in 2014 there were 257 instances of a transition from a runner on first base with nobody out to a state with nobody on base with nobody out. Likely, the vast majority of these transitions were due to a two-run home run, but perhaps a stolen base or pickoff attempt together with a grossly errant throw which allowed the baserunner to score could have happened a few times. Since there were 10,951 instances of a runner on first base with nobody out, the transition probability from (runner on first with nobody out) to (nobody on with nobody out) we model as 257/10,951 in the 2014 season.

We add an extra state to the Markov chain called the “end of inning” state, which acts as an absorbing state. In such, there are 25 states in this Markov chain. Let P denote the 25 × 25 transition matrix generated using MLB data. If PT represents the 24 × 24 transition matrix which is the same as P but with the row and column representing the end of inning state (the absorbing state) removed, then it is a standard result (see, for instance, Sheldon Ross’s book7) that Sij, the expected number of visits to state j starting from state i, is given in matrix form by:

S=(I–PT)–1

We can also define an expected number of runs produced for each transition by looking at MLB data. The number of runs produced for each transition were recorded, then divided by the total number of transitions to get an expected number of runs generated per transition. For instance, in 2014 there were 133 runs generated by transitions from (runner on first with nobody out) to (runner on second with nobody out). There were 823 such transitions in that season, so the expected number of runs per transition is 0.1616. (Here, most of the time the transition was made by the runner on first stealing second, but occasionally the transition was made by the batter hitting a double in which the runner scored from first.) Let R denote the matrix whose entries rij are the expected number of runs per transition from state i to state j. Then the vector r, with ri= ΣjRij, will give us the expected number of runs scored by transitioning out of state i. The product Sr will then give us the expected number of further runs scored in a half inning from each base running situation under this model. We can use this to evaluate the wisdom of making the first or last out at third base using some of the metrics introduced in the last section.

We can analyze the threshold values for sending a runner from second to third base in the same way as we did before. A summary of the data for the 2014 season appears in Table 9. We notice that the threshold values for sending a runner to third base in the absence of a runner on first are higher than the threshold for sending a runner to second when there are zero or two outs, but lower when there is one out. Also, the Markov model has threshold values in line with those generated using the Baseball Prospectus data.

Table 9

advance to 0 outs 1 out 2 outs
2nd base 0.704 0.717 0.689
3rd base (lone runner) 0.791 0.663 0.839
3rd base (also runner on 1st) 0.715 0.738 0.902
home (lone runner) 0.888 0.730 0.320
home (runner on 1st) 0.950 0.706 0.364
home (runner on 2nd) 0.891 0.794 0.387
home (runner on 1st and 2nd) 0.904 0.777 0.478

Threshold Values for sending a runner in various situations under the Markov chain model, using 2014 MLB data.

Upon closer inspection we see that it would actually be more damaging to make the first out at home plate. The threshold value here of 0.888 (as a lone runner) indicates that we really don’t want to make the first out at home plate. Also, the fact that the threshold value for sending a runner to third in the presence of a runner at first with one out (0.738) is greater than the threshold for sending a runner to second with one out (0.717) is not consistent that there is anything “special” about third base with zero or two outs which would allow us to be less careful about making the second out at third base.

In order to verify the consistency of this model, we did the procedure indicated above for the 2013 MLB season. That is, we examined all of the play-by-play data to build a transition matrix for that season, examined the runs scored for each transition in that season, and calculated the threshold values based on those. The results are shown in Table 10. In 2013, we do see that sending a runner to third base with one out seems less risky than sending a runner to second with one out, while this is not the case with zero or two outs. However, we also note that the situation which requires the highest threshold is still sending a runner home with zero outs.

Table 10

Advance to 0 outs 1 out 2 outs
2nd base 0.686 0.730 0.712
3rd base (lone runner) 0.780 0.653 0.859
3rd base (also runner on 1st) 0.748 0.715 0.843
home (lone runner) 0.886 0.726 0.321
home (runner on 1st) 0.919 0.730 0.406
home (runner on 2nd) 0.916 0.788 0.419
home (runner on 1st and 2nd) 0.853 0.777 0.515

Threshold Values for sending a runner in various situations under the Markov chain model, using 2013 MLB data.

Again using the Markov chain model as a guide, we formulate separate Markov transition matrices for each MLB team based on 2014 play-by-play data. The number of runs scored per transition were also sorted per team in order to calculate all of the values above for each team in 2014. The data here do not always support the adage nearly as well.

As we did in the first section, we can examine the threshold for advancing to third base. We first look to see whether the threshold for advancing to third base is lower with one out than with zero or two outs. For two of 30 teams (Baltimore and Boston), the thresholds for going to third with one out was higher than that with zero outs with no other runners on base. For four teams (Oakland, San Diego, Cleveland, and Chicago White Sox), the threshold for going to third with one out was higher than that with two outs with no other runners on base. When there is an additional runner on first base, the situation is even less consistent. For seven of the 30 teams the threshold for advancing to third with one out eclipses that with two outs and for 17 of 30 teams the threshold for advancement with one out exceeds that with zero outs when there is another runner on first base. Only 10 of the 30 teams had all four of these inequalities in a direction consistent with the adage.

Finally, we calculate all of the thresholds for each team to see which are the highest. Of the 30 teams in 2014, 27 had their highest thresholds for either sending a runner home with zero outs (with various configurations of baserunners on the other bases) or sending a runner to third base with two outs in the presence of a runner on first base. One team for which these advances did not have the highest threshold using the Markov model is the Kansas City Royals. Their highest thresholds were sending a runner to third with nobody else on base with zero outs and with two outs. In fact, both of these thresholds are greater than one, indicating that the 2014 Royals should never have a runner try to stretch a double into a triple with zero or two outs, even if success is certain. The other two teams are the Cincinnati Reds and the Baltimore Orioles, which had the highest threshold in sending a runner home with one out in the presence of two other baserunners. Putting this together, we can say the following:

Never make the first out at home. Don’t make the last out at third base if there is another runner behind you.

CONCLUSIONS AND DISCUSSION

We have examined a few ways of using data to support or refute a popular adage.

Through the computation of threshold values for advancing to each base given the current configuration of runners, we initially find that, in general, advancing to third base with one out has a lower threshold than doing so with zero or two outs (though this is not true of all data sets with all runner configurations). This seems to suggest that making the second out at third base is less damaging than the first or last. However, the threshold for advancing to third with one out is usually close to that with zero outs, so it is not immediately clear that there is really anything special about making the first out at third base.

We also examined the expected number of runs forfeited by unsuccessful advance. With this, we saw that we are poised to do the most damage by unsuccessfully advancing a runner from second to third with 0 outs. By analyzing this, it seems odd to recommend that we never make the last out at third base; doing so simply doesn’t squander as many expected runs as we would by making the first or second out there.

If all we need is one run in an inning, the threshold for advancing to third base is much higher with two outs than with zero or one. If more than one run is needed, then the thresholds for advancing to third base are all pretty high.

We also looked at other bases to see if third base is special enough to warrant such an adage. We find the highest thresholds to occur when advancing to third with two outs, advancing to home with zero outs, and advancing to home anytime more than one run is needed in an inning.

By generalizing this analysis to a Markov chain model, we reach similar conclusions. However, when using team-by-team data rather than MLB data as a whole, we find large variation among the threshold values. Even so, the vast majority of the teams seem to be best served with the updated adage:

Never make the last out at third base. Never make the first out at home plate. And never make any out at home plate if more than one run is needed in the inning.

The evidence that making the first out at third base is substantially worse than other situations is just not that strong. We’ve presented a few arguments for it, but all seem like cherry-picking the data that support the argument. Because there is no “right” or “wrong” way to analyze data which are relevant to the adage, the best we can do is to use data to help make our point. Doesn’t this seem to be the case with any baseball argument that one thing/player/strategy is better than another?

Finally, it should be noted that all of these analyses were done using Major League Baseball data. Whether the adage provides wisdom backed with data for Little Leaguers or recreational baseball leagues remains to be seen.

RYAN GANTNER has a Ph.D. in Mathematics from the University of Minnesota and teaches mathematics at St. John Fisher College in Rochester, New York. Like most projects of his, this one developed as an offshoot of a student research project. He is new to SABR but hopes to guide students in this direction in the future.

Notes

1 “Baseball Prospectus/Statistics/Custom Statistics Reports: Run Expectations,” accessed November 20, 2015, http://www.baseballprospectus.com/sortable/index.php?cid=1657937.

2 David W. Smith, “Are outs on the bases more harmful than other types of outs?” accessed December 30, 2015, http://www.retrosheet.org/Research/SmithD/OutsOnBases.pdf

3 Zachary Levine, “Don’t make the last out at third (The first is forgivable),” Houston Chronicle, February 14, 2009, http://blog.chron.com/unofficialscorer/2009/02/dont-make-the-last-out-at-third-the-first-is-forgivable

4 Zachary Levine, “More on the last out at third base; goodbye to a friend,” Houston Chronicle, February 16, 2009, http://blog.chron.com/unofficialscorer/2009/02/more-on-the-last-out-at-third-base-goodbye-to-a-friend.

5 “Retrosheet Event Files,” accessed October 21, 2015, http://www.retrosheet.org/game.htm.

6 Mark D. Pankin, “Baseball as a Markov Chain,” in The Great American Baseball Stat Book, ed. Bill James (New York: Ballantine Books, 1987), 520–24.

7 Sheldon M. Ross, Introduction to Probability Models, Tenth Edition, (Burlington, MA: Academic Press, 2009).