To calculate the expected win ratio based on the runs scored and runs allowed for the Cincinnati Reds in the 2021 season, we will use the given equation for the Win Ratio.
Step 1: Calculate the Expected Win Ratio for the Reds
Using the formula: \[ \text{Win Ratio} = \frac{(\text{runs scored})^2}{(\text{runs scored})^2 + (\text{runs allowed})^2} \]
For the Reds:
- Runs Scored = 786
- Runs Allowed = 760
Calculating the squared values:
- \((\text{Runs Scored})^2 = 786^2 = 616596\)
- \((\text{Runs Allowed})^2 = 760^2 = 577600\)
Now plug these values into the Win Ratio formula: \[ \text{Win Ratio} = \frac{616596}{616596 + 577600} \] \[ = \frac{616596}{1199196} \] \[ = 0.514 \]
Rounding to the nearest thousandth gives us 0.514.
Step 2: Calculate Expected Wins for the Reds
To calculate the expected wins: \[ \text{Expected Wins} = \text{Win Ratio} \times \text{Total Games} \]
For a 162-game season: \[ \text{Expected Wins} = 0.514 \times 162 \approx 83.508 \] Rounding to the nearest whole number gives us 84 expected wins.
Step 3: Calculate Expected Wins for a Different Team (794 runs scored and 673 runs allowed)
Using the same Win Ratio formula for the new team:
- Runs Scored = 794
- Runs Allowed = 673
Calculating the squared values:
- \((\text{Runs Scored})^2 = 794^2 = 630036\)
- \((\text{Runs Allowed})^2 = 673^2 = 452929\)
Now plug these values into the formula: \[ \text{Win Ratio} = \frac{630036}{630036 + 452929} \] \[ = \frac{630036}{1087965} \approx 0.578 \]
Now calculate the expected wins: \[ \text{Expected Wins} = 0.578 \times 162 \approx 93.576 \] Rounding to the nearest whole number gives us 94 expected wins for the team that scored 794 runs and allowed 673 runs.
Summary of Results
- Expected Win Ratio for the Reds: 0.514
- Expected Wins for the Reds: 84
- Expected Wins for the new team: 94
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