No sport generates as many statistics as baseball. Reporters, managers, and fans

Question

No sport generates as many statistics as baseball. Reporters, managers, and fans argue and discuss strategies on the basis of these statistics. An article in Chance (“A Statistician reads the sports page,” Hal S. Stem, Vol 1, Winter 1997) offers baseball lovers another opportunity to analyze numbers associated with the game. Table 1 lists the probabilities of scoring at least one run in situations that are defined by the number of outs and bases occupied. For example, the probability of scoring at least one run when there are no outs and a man is on first base is .39. If the bases are loaded with one out, then the probability of scoring any runs is .67. (Probabilities are based on results from the American League during 1989 season. The results for the National League are also shown in the article and are similar.)

Table 1 Probability of Scoring Any Runs

Bases                                                               0                      1                      2

Occupied                                                         Outs                 Out                  Outs    

Bases Empty                                                    .26                   .16                   .07

First Base                                                        .39                   .26                   .13

Second Base                                                    .57                   .42                   .24

Third Base                                                       .72                   .55                   .28

First base and Second Base                            .59                   .45                   .24

First base and third base                                .76                   .61                   .37

Second base and third base                            .83                   .74                   .37

Bases Loaded                                                  .81                   .67                   .43

Table 1 allows us to determine the best strategy in a variety of circumstances. This case will concentrate on the strategy of the sacrifice bunt. The purpose of the sacrifice bunt is to sacrifice the batter to move base runners to the next base. It can be employed when there are fewer than two outs and men on base. Ignoring the suicide squeeze, any of four outcomes can occur:

The bunt is successful. The runner (or runners) advances one base, and the batter is out.

The batter is out but fails to advance the runner.

The batter bunts into a double play.

The batter is a safe (hit or error), and the runner advances.

Suppose that you are and American league manager. The game is tied in the middle innings of a game, and there is a runner on the first base with no one out. Given the following probabilities of the four outcomes of a bunt for the batter at the plate, should you signal the batter to sacrifice bunt?

P (Outcome 1) = 0.75

P (Outcome 2) = .10

P (Outcome 3) = .10

P (Outcome 4) = .05

Assume for simplicity that after the hit or error in outcome 4, there will be men on first and second base and no one out.

Explanation / Answer

If we have outcome 1, the probability of scoring a run is 0.42

If we have outcome 2, the probability of scoring a run is 0.26

If we have outcome 3, the probability of scoring a run is 0.07

If we have outcome 4, the probability of scoring a run is 0.59

so the probability of scoring a run if the batter bunts is, (we use the probabbility of each outcome)

0.75*0.42 + 0.10*0.26 + 0.10*0.07 +0.05*0.59 = 0.3775

Now the probability of scoring with the first base loaded and 0 outs is 0.39

As this is more than the probability of scoring a run if the batter bunts, it will be unadvisable to signal the batter to bunt.

6.3)

Considerin 0 outs

The probability of scoring a run with second base loaded with zero outs is 0.57.

The probability of scoring a run with no bases loaded with 1 out is 0.16.

So the probability of scoring a run after the steal is,

0.68*0.57 + 0.32*0.16 = 0.4388

Since the probability of scoring a run with first base loaded and 0 outs is 0.39

it will be advisable to steal a base.

Considerin 1 out

The probability of scoring a run with 2nd base loaded with 1 outs is 0.42

The probability of scoring a run with no bases loaded with 2 outs is 0.07

So the probability of scoring a run after the steal is,

0.68*0.42 + 0.32*0.07 = 0.308

Since the probability of scoring a run with first base loaded and 1 out is 0.26

it will be advisable to steal a base.

Considerin 2 outs

The probability of scoring a run with 2nd base loaded with 2 outs is 0.24

The probability of scoring a run with 0 bases loaded with 3 outs is 0

So the probability of scoring a run after the steal is,

0.68*0.24 + 0.32*0 = 0.1632

Since the probability of scoring a run with first base loaded and 2 outs is 0.13

it will be advisable to steal a base.

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