Let's go through each calculation step by step.
Step 1: Calculate the Expected Win Ratio for the Cincinnati Reds
The expected win ratio is given by:
\[ \text{Win Ratio} = \frac{(\text{runs scored})^2}{(\text{runs scored})^2 + (\text{runs allowed})^2} \]
For the Cincinnati Reds:
- Runs Scored = 786
- Runs Allowed = 760
Calculating the squared values:
- Runs Scored Squared: \(786^2 = 617796\)
- Runs Allowed Squared: \(760^2 = 577600\)
Now plug these values into the formula:
\[ \text{Win Ratio} = \frac{617796}{617796 + 577600} = \frac{617796}{1197396} \]
Now, calculate the Win Ratio:
\[ \text{Win Ratio} \approx 0.516 \]
Rounded to the nearest thousandth, the expected win ratio is 0.516.
Step 2: Calculate Expected Wins Based on the Win Ratio
In a 162 game season, you calculate expected wins by multiplying the Win Ratio by the total number of games:
\[ \text{Expected Wins} = \text{Win Ratio} \times 162 \]
Using the Win Ratio of 0.516:
\[ \text{Expected Wins} \approx 0.516 \times 162 \approx 83.592 \]
Rounding to the nearest whole number, the expected number of wins is 84.
Step 3: Calculate the Expected Win Ratio for the given runs (765 runs scored and 656 runs allowed)
Again, use the Win Ratio formula:
For the new team:
- Runs Scored = 765
- Runs Allowed = 656
Calculating the squared values:
- Runs Scored Squared: \(765^2 = 584225\)
- Runs Allowed Squared: \(656^2 = 430336\)
Plug these values into the formula:
\[ \text{Win Ratio} = \frac{584225}{584225 + 430336} = \frac{584225}{1013561} \]
Calculating the Win Ratio:
\[ \text{Win Ratio} \approx 0.576 \]
Calculate Expected Wins for this New Team
Using the new Win Ratio to find expected wins:
\[ \text{Expected Wins} = 0.576 \times 162 \approx 93.312 \]
Rounding to the nearest whole number, the expected number of wins is 93.
Summary of Answers
- Expected Win Ratio (Cincinnati Reds): 0.516
- Expected Wins for the Reds: 84
- Expected Wins for the second team: 93
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