To compute the mean number of home runs per game and then use the Poisson distribution to estimate probabilities for various outcomes (such as no home runs, two home runs, and at least four home runs), follow these steps:

Step 1: Find the Mean Number of Home Runs per Team for 2022

• You mentioned a dataset, "Baseball 2022," but since I don't have direct access to the Excel file, you need to provide either the total number of home runs hit across all teams in the season or upload the data file.

If you provide that, I can compute the following:

1. Mean number of home runs per team: $\text{Mean home runs per team} = \frac{\text{Total number of home runs in 2022}}{\text{Number of teams}}$

2. Mean home runs per game: $\lambda = \frac{\text{Mean home runs per team}}{162} \times 2$ (since each team plays 162 games and there are 2 teams in each game).

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Step 2: Apply the Poisson Distribution

The Poisson distribution formula for the probability of $k$ home runs in a game is given by: $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$ Where:

• $\lambda$ is the mean number of home runs per game.
• $k$ is the number of home runs we are calculating the probability for.

(a) Find the probability that there are no home runs in a game ($k = 0$):

$P(X = 0) = \frac{\lambda^0 e^{-\lambda}}{0!} = e^{-\lambda}$

(b) Find the probability that there are two home runs in a game ($k = 2$):

$P(X = 2) = \frac{\lambda^2 e^{-\lambda}}{2!}$

(c) Find the probability that there are at least four home runs in a game ($k \geq 4$):

This is the complement of the probability that there are fewer than four home runs: $P(X \geq 4) = 1 - P(X < 4)$ Where: $P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

To proceed with exact numbers, please upload the data file or provide the total number of home runs across all teams in the 2022 season. I will then compute the mean $\lambda$ and use it to determine the probabilities.

Would you like to upload the file or provide the home run totals?

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Relative Questions:

1. How is the Poisson distribution different from the binomial distribution?
2. Why do we multiply by 2 to account for both teams in the game?
3. How is the mean of a Poisson distribution related to its variance?
4. What are some real-world examples, other than baseball, where Poisson distribution is used?
5. How does increasing $\lambda$ affect the shape of the Poisson distribution?

Tip: In Poisson distribution problems, the mean and variance are equal, which simplifies certain calculations.