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Sagot :
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.
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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