To compute the mean number of home runs per game for Major League Baseball in 2022, we'll follow the steps you've outlined.

Step 1: Calculate the Mean Home Runs per Team

1. Total Home Runs in 2022: First, we need the total number of home runs hit in the 2022 season. For this, let's say we found that total home runs were approximately 5,200.

2. Number of Teams: There are 30 teams in Major League Baseball.

3. Mean Home Runs per Team: $\text{Mean Home Runs per Team} = \frac{\text{Total Home Runs}}{\text{Number of Teams}} = \frac{5200}{30} \approx 173.33$

Step 2: Mean Home Runs per Game

1. Divide by Games: Each team plays 162 games. To find the mean home runs per game per team: $\text{Mean Home Runs per Team per Game} = \frac{173.33}{162} \approx 1.07$

2. Multiply by 2 (for two teams): $\text{Mean Home Runs per Game} = 1.07 \times 2 \approx 2.14$

Step 3: Estimate Using Poisson Distribution

To use the Poisson distribution, we can let $\lambda$ (the mean number of home runs per game) be approximately 2.14.

The Poisson probability mass function is: $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$

Where $P(X=k)$ is the probability of $k$ home runs in a game, $\lambda$ is the average rate (2.14), and $k$ is the number of home runs.

Summary

• Mean Home Runs per Game: Approximately 2.14.

If you need further details on how to compute probabilities for specific numbers of home runs using the Poisson distribution or any other questions, feel free to ask!

Questions for Further Exploration:

1. What are the probabilities of hitting 0, 1, 2, or more home runs in a game using the Poisson distribution?
2. How does the mean number of home runs per game compare to previous seasons?
3. What factors could influence the number of home runs in a game?
4. How can we use historical data to predict future home run trends?
5. Are there any teams known for having higher home run averages?

Tip:

Always check multiple seasons of data to identify trends in home runs or other statistics for better predictions.