To calculate the expected win ratio based on the provided formula, we can substitute the figures from the Cincinnati Reds for the 2021 season into the win ratio equation.
The formula for the Win Ratio is:
\[ \text{Win Ratio} = \frac{(\text{runs scored})^2}{(\text{runs scored})^2 + (\text{runs allowed})^2} \]
Using the Reds' runs scored (786) and runs allowed (760):
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Calculate \((\text{runs scored})^2\): \[ (786)^2 = 616996 \]
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Calculate \((\text{runs allowed})^2\): \[ (760)^2 = 577600 \]
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Compute the total of \((\text{runs scored})^2 + (\text{runs allowed})^2\): \[ 616996 + 577600 = 1192596 \]
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Now, substitute these values back into the Win Ratio formula: \[ \text{Win Ratio} = \frac{616996}{1192596} \]
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Perform the division: \[ \text{Win Ratio} \approx 0.517 \]
So, rounded to the nearest thousandth, the expected win ratio is 0.517.
Next, we calculate the expected number of wins in a 162 game season:
\[ \text{Expected Wins} = \text{Win Ratio} \times 162 \]
Substituting the win ratio we calculated:
\[ \text{Expected Wins} = 0.517 \times 162 \approx 83.754 \]
Rounding to the nearest whole number, the expected number of wins for the Reds would be 84 wins.
In summary:
- The expected win ratio is 0.517.
- The expected number of wins in a 162 game season is 84 wins.