To find the expected win ratio using the formula given, we can substitute the runs scored and runs allowed into the equation.
Step 1: Calculate the Win Ratio for the Cincinnati Reds (2021)
Given:
- Runs Scored = 786
- Runs Allowed = 760
Win Ratio is calculated as follows:
\[ \text{Win Ratio} = \frac{(\text{runs scored})^2}{(\text{runs scored})^2 + (\text{runs allowed})^2} \]
Substituting the values:
\[ \text{Win Ratio} = \frac{786^2}{786^2 + 760^2} \]
First, calculate \(786^2\) and \(760^2\):
\[ 786^2 = 617796 \] \[ 760^2 = 577600 \]
Now plug these values back into the equation:
\[ \text{Win Ratio} = \frac{617796}{617796 + 577600} = \frac{617796}{1197396} \]
Now perform the division:
\[ \text{Win Ratio} \approx 0.516 \]
Rounding to the nearest thousandth:
\[ \text{Win Ratio} \approx 0.516 \]
Step 2: Calculate Expected Wins for the Cincinnati Reds
In a 162 game season, the expected number of wins can be calculated by multiplying the win ratio by the total number of games:
\[ \text{Expected Wins} = \text{Win Ratio} \times 162 \] \[ \text{Expected Wins} = 0.516 \times 162 \approx 83.592 \]
Rounding to the nearest whole number gives:
\[ \text{Expected Wins} \approx 84 \]
Step 3: Calculate Expected Wins for the Team that Scored 781 Runs and Allowed 696 Runs
Given:
- Runs Scored = 781
- Runs Allowed = 696
Using the same Win Ratio formula:
\[ \text{Win Ratio} = \frac{781^2}{781^2 + 696^2} \]
Calculate \(781^2\) and \(696^2\):
\[ 781^2 = 610561 \] \[ 696^2 = 485616 \]
Now substitute these values into the Win Ratio formula:
\[ \text{Win Ratio} = \frac{610561}{610561 + 485616} = \frac{610561}{1093177} \]
Now perform the division:
\[ \text{Win Ratio} \approx 0.558 \]
Final Calculation: Expected Wins for this Team
To find the expected wins in a 162 game season:
\[ \text{Expected Wins} = \text{Win Ratio} \times 162 \] \[ \text{Expected Wins} = 0.558 \times 162 \approx 90.396 \]
Rounding to the nearest whole number gives:
\[ \text{Expected Wins} \approx 90 \]
Summary
- The expected win ratio for the Cincinnati Reds is 0.516.
- The expected number of wins for the Cincinnati Reds is 84.
- The expected number of wins for the team scoring 781 runs and allowing 696 runs is 90.