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Ron coaches a baseball team. There are three innings left in the game, and the team is losing by four runs. Ron is trying to decide whether to replace the pitcher or keep the pitcher in for another inning. In the past, when losing by four runs, he has replaced the pitcher a total of 14 times and kept the pitcher in a total of 10 times. The table shows the results of those decisions at the end of nine innings:

\begin{tabular}{|l|c|c|}
\hline & Replaced pitcher & Kept pitcher \\
\hline Won game & 8 & 4 \\
\hline Lost game & 4 & 5 \\
\hline Tied game & 2 & 1 \\
\hline Total & 14 & 10 \\
\hline
\end{tabular}

Based on the information in the table, if the goal is to win the game in nine innings, should Ron replace the pitcher or keep the pitcher in?

A. Ron replacing or not replacing the pitcher has no advantage.
B. Ron should replace the pitcher with a relief pitcher.
C. Ron should not replace the pitcher with a relief pitcher.
D. There is not enough information to determine if Ron should replace the pitcher or not.

Sagot :

GinnyAnswer
Let's analyze the problem step by step to determine the best course of action for Ron, based on the given data.

We have two scenarios to consider:
1. Replacing the pitcher
2. Keeping the pitcher

### Step 1: Data Compilation
Replaced pitcher:
- Won: 8
- Lost: 4
- Tied: 2
- Total games: 14

Kept pitcher:
- Won: 4
- Lost: 5
- Tied: 1
- Total games: 10

### Step 2: Calculate the Win Probability for Each Scenario
Probability of winning when the pitcher is replaced:
[tex]\[ \text{Win Probability (Replaced)} = \frac{\text{Number of Wins when Replaced}}{\text{Total Games when Replaced}} \][/tex]
[tex]\[ \text{Win Probability (Replaced)} = \frac{8}{14} \][/tex]
[tex]\[ \text{Win Probability (Replaced)} = \frac{4}{7} \][/tex]
[tex]\[ \text{Win Probability (Replaced)} \approx 0.5714 \][/tex]

Probability of winning when the pitcher is kept:
[tex]\[ \text{Win Probability (Kept)} = \frac{\text{Number of Wins when Kept}}{\text{Total Games when Kept}} \][/tex]
[tex]\[ \text{Win Probability (Kept)} = \frac{4}{10} \][/tex]
[tex]\[ \text{Win Probability (Kept)} = 0.4 \][/tex]

### Step 3: Determine the Best Decision
- The win probability when replacing the pitcher is approximately 0.5714.
- The win probability when keeping the pitcher is 0.4.

Since 0.5714 (57.14%) is greater than 0.4 (40%), replacing the pitcher gives a higher probability of winning.

### Conclusion
The analysis shows that replacing the pitcher has a higher win probability compared to keeping the pitcher in. Thus, the best decision would be:

B. Ron should replace the pitcher with a relief pitcher.
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