To calculate the expected win percentage (win ratio) using the formula given:
\[ \text{Win Ratio} = \frac{(r_{\text{scored}})^2}{(r_{\text{scored}})^2 + (r_{\text{allowed}})^2} \]
For the Cincinnati Reds in 2021:
- Runs scored (\(r_{\text{scored}}\)) = 786
- Runs allowed (\(r_{\text{allowed}}\)) = 760
Plugging in the values:
\[ \text{Win Ratio} = \frac{(786)^2}{(786)^2 + (760)^2} \]
Calculating \( (786)^2 \) and \( (760)^2 \):
\[ (786)^2 = 617,796 \] \[ (760)^2 = 577,600 \]
Now plugging these back into the equation:
\[ \text{Win Ratio} = \frac{617,796}{617,796 + 577,600} = \frac{617,796}{1,195,396} \]
Calculating the division:
\[ \text{Win Ratio} \approx 0.517 \]
Rounding to the nearest thousandth, the Win Ratio is:
\[ \text{Win Ratio} = 0.517 \]
Now to find how many wins this would translate to in a 162-game season:
\[ \text{Expected Wins} = \text{Win Ratio} \times \text{Total Games} = 0.517 \times 162 \]
Calculating this gives:
\[ \text{Expected Wins} \approx 83.754 \]
Rounding to the nearest whole number, this gives:
\[ \text{Expected Wins} = 84 \]
Next, for a different team that scored 765 runs and allowed 658 runs:
Using the same formula:
- Runs scored (\(r_{\text{scored}}\)) = 765
- Runs allowed (\(r_{\text{allowed}}\)) = 658
\[ \text{Win Ratio} = \frac{(765)^2}{(765)^2 + (658)^2} \]
Calculating \( (765)^2 \) and \( (658)^2 \):
\[ (765)^2 = 585,225 \] \[ (658)^2 = 432,964 \]
Now plugging these back into the equation:
\[ \text{Win Ratio} = \frac{585,225}{585,225 + 432,964} = \frac{585,225}{1,018,189} \]
Calculating the division:
\[ \text{Win Ratio} \approx 0.574 \]
For the expected wins based on this Win Ratio in a 162-game season:
\[ \text{Expected Wins} = \text{Win Ratio} \times \text{Total Games} = 0.574 \times 162 \]
Calculating this gives:
\[ \text{Expected Wins} \approx 93.108 \]
Rounding to the nearest whole number:
\[ \text{Expected Wins} = 93 \]
Final Results:
- The expected win ratio for the Cincinnati Reds: 0.517
- Expected wins for the Reds: 84
- Expected wins for the second team: 93