For Major League Baseball the expected win percentage is given by

Win Ratio =(runs scored)²+ (runs allowed)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 765 runs and allowed 658 runs? Round to the nearest whole number.

1 answer

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:

  1. The expected win ratio for the Cincinnati Reds: 0.517
  2. Expected wins for the Reds: 84
  3. Expected wins for the second team: 93