Question

For Major League Baseball the expected win percentage is given by



Win Ratio 
=
 
(runs scored)²
(runs scored)²  (runs allowed)²
Win Ratio = 
(runs scored)²  (runs allowed)²
(runs scored)²
​




For the 2021 season the Cincinnati Reds scored 786 runs and allowed 760 runs.



Based on the Win Ratio equation what is the expected win ratio to the nearest thousandth?



(Note that the thousandth is the 3rd digit after the decimal so a number such as 0.4279 would be rounded to 0.428.)



In a 162 game season how many wins would this have given the Reds (their real record was 83-79)? Round to the nearest whole number.




 



In a 162 game season how many wins would you expect a team to win that scored
781
781 runs and allowed
696
696 runs?  Round to the nearest whole number.

Answer 1

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
1. The expected win ratio for the Cincinnati Reds is **0.516**.
2. The expected number of wins for the Cincinnati Reds is **84**.
3. The expected number of wins for the team scoring 781 runs and allowing 696 runs is **90**.