Case 6.2 Case 6.3. (The first sentence below is actually: Refer to Case 6.2.) Hi

Question

Case 6.2

Case 6.3. (The first sentence below is actually: Refer to Case 6.2.)

Hi I would like to request the step-by-step solutioin to the the following Problem:

Statistics for Management and Economics 10th Edition: Chapter 6, Case 6.3 "Should He Attempt to Steal a Base" (and possibly to Case 6.2).

I was also wandering what would be total cost of that solution. For one case and for all the cases.

I would appeciate your help on that matter. Thank you.

Cheers,

Pawel

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. Stern, 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 the 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. lf the bases are loaded with one out, (Case 6.2 continued) then the probability of scoring any runs is 67 (Probabilities are based on results from the American League during the 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 occupied O Outsi Out 2 Outs .26 16 .07 Bases empty 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 offour outcomes can Occur:

Explanation / Answer

6.2)

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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