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 719 719 runs and allowed 619 619 runs? Round to the nearest whole number.

Answer 1

To begin, we'll calculate the expected win ratio for the Cincinnati Reds for the 2021 season using the formula provided:

\[ \text{Win Ratio} = \frac{(\text{runs scored})^2}{(\text{runs scored})^2 + (\text{runs allowed})^2} \]

The Reds scored 786 runs and allowed 760 runs. Plugging these values into the formula:

1. Calculate the runs scored squared:

\[ (786)^2 = 617796 \]

2. Calculate the runs allowed squared:

\[ (760)^2 = 577600 \]

3. Now, calculate the sum of squared runs:

\[ (786)^2 + (760)^2 = 617796 + 577600 = 1196396 \]

4. Finally, calculate the win ratio:

\[ \text{Win Ratio} = \frac{617796}{1196396} \]

5. Performing the division:

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

Rounding to the nearest thousandth gives us:

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

Next, to find the expected number of wins in a 162-game season, we multiply the win ratio by the total number of games:

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

Calculating that:

\[ \text{Expected Wins} \approx 83.592 \]

Rounding to the nearest whole number, the expected number of wins for the Reds would be:

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

Now, let's calculate the expected win ratio for a team that scored 719 runs and allowed 619 runs. Using the same formula:

1. Calculate the runs scored squared:

\[ (719)^2 = 517761 \]

2. Calculate the runs allowed squared:

\[ (619)^2 = 383761 \]

3. Calculate the sum of squared runs:

\[ (719)^2 + (619)^2 = 517761 + 383761 = 901522 \]

4. Now calculate the win ratio:

\[ \text{Win Ratio} = \frac{517761}{901522} \]

5. Performing the division:

\[ \text{Win Ratio} \approx 0.574 \]

To find the expected wins for the second case in a 162-game season:

\[ \text{Expected Wins} = 0.574 \times 162 \]

Calculating that gives:

\[ \text{Expected Wins} \approx 93.108 \]

Rounding to the nearest whole number, the expected number of wins would be:

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

To summarize:

• The expected win ratio for the Cincinnati Reds: 0.516
• Expected wins for the Reds: 84
• Expected wins for the team scoring 719 runs and allowing 619 runs: 93

Answer 2

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Answer 3

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