For major league baseball that expanded win percentage is given by win ratio equals (runs scored) to the power of two over (run scored) the power of 2+ runs allowed to the power of two for the 2021 season the Cincinnati Reds scored 786 runs and allowed 760 runs based on the wind ratio equation. What is the expected ratio to the nearest thousand note the thousand is the third digit after the decimal number such as 0.4279 would be rounded to 0.428.

Question

in a 162 game season how many wins would this have given the Reds (their real record record was 83-79 round the nearest whole number

N a 162 game season how many wins would you expect a team to win that scored 774 runs and allowed 666 runs round the nearest whole number

Answer 1

To calculate the expected win percentage using the expanded win percentage formula, we will use the following formula:

\[ \text{Win Ratio} = \frac{(\text{Runs Scored})^2}{(\text{Runs Scored})^2 + (\text{Runs Allowed})^2} \]

1. Calculate for the Cincinnati Reds with 786 Runs Scored and 760 Runs Allowed.

First, plug in the values for the Reds:

\[ \text{Runs Scored} = 786 \] \[ \text{Runs Allowed} = 760 \]

Calculating the squared values:

\[ (\text{Runs Scored})^2 = 786^2 = 617796 \] \[ (\text{Runs Allowed})^2 = 760^2 = 577600 \]

Now, substitute these values into the win ratio formula:

\[ \text{Win Ratio} = \frac{617796}{617796 + 577600} = \frac{617796}{1197396} \]

Calculating this gives:

\[ \text{Win Ratio} \approx 0.5164 \]

Rounding this to the nearest thousandth yields:

\[ \text{Win Ratio} \approx 0.516 \]

Now calculate expected wins in a 162-game season:

To find the expected number of wins:

\[ \text{Expected Wins} = \text{Win Ratio} \times \text{Total Games} \] \[ \text{Expected Wins} = 0.516 \times 162 \approx 83.592 \]

Rounding to the nearest whole number gives us:

\[ \text{Expected Wins} \approx 84 \]

2. For a Team that Scored 774 Runs and Allowed 666 Runs:

Using the same win ratio formula:

\[ \text{Runs Scored} = 774 \] \[ \text{Runs Allowed} = 666 \]

Calculating the squared values:

\[ (\text{Runs Scored})^2 = 774^2 = 599076 \] \[ (\text{Runs Allowed})^2 = 666^2 = 443556 \]

Now, substitute into the win ratio formula:

\[ \text{Win Ratio} = \frac{599076}{599076 + 443556} = \frac{599076}{1045632} \]

Calculating this gives:

\[ \text{Win Ratio} \approx 0.5725 \]

Rounding to the nearest thousandth:

\[ \text{Win Ratio} \approx 0.573 \]

Expected Wins in a 162-game Season:

Calculating expected wins:

\[ \text{Expected Wins} = 0.573 \times 162 \approx 92.754 \]

Rounding to the nearest whole number gives us:

\[ \text{Expected Wins} \approx 93 \]

Summary:

• Expected Win Ratio for Reds: 0.516
• Expected Wins for Reds: 84
• Expected Win Ratio for a team with 774 runs scored and 666 runs allowed: 0.573
• Expected Wins for that team: 93