angela5210
TO BUNT OR NOT TO BUNT, THAT IS THE 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. 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. If the bases are loaded with one out, the probability of scoring any runs is .67.
Table 1: Probability of Scoring Any Runs
Bases Occupied 0 Outs 1 Out 2 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
(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 allows us to determine the best strategy in a variety of circumstances. This case will concentrate on the strategy of the sacrifice
bunt. The purposed 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:
1. The bunt is successful. The runner (or runners) advances one base, and the batter is out.
2. The batter is out but fails to advance the runner.
3. The batter bunts into a double play.
4. The batter is safe (hit or error), and the runner advances.
Suppose that you are an American League manager. The game is tied in the middle innings of a
game, and there is a runner on 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) = .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.