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Let's break down and solve the problem based on the information provided in the image.

You are given a Poisson distribution with \( \lambda = 4.7 \), and you need to calculate the following probabilities:

### Part (a): Probability of exactly 5 occurrences
For a Poisson distribution, the probability of exactly \( k \) occurrences is given by:

\[
P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
\]

For \( k = 5 \) and \( \lambda = 4.7 \), we can calculate this value.

### Part (b): Probability of more than 6 occurrences
This is the cumulative probability for \( X > 6 \), which can be found by subtracting the cumulative probability for \( X \leq 6 \) from 1:

\[
P(X > 6) = 1 - P(X \leq 6)
\]

### Part (c): Probability of 3 or fewer occurrences
This is the cumulative probability for \( X \leq 3 \), which can be calculated using:

\[
P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
\]

I will now compute these probabilities for you.Here are the calculated probabilities:

- **Part (a)**: The probability of exactly 5 occurrences is approximately \( 0.1738 \).
- **Part (b)**: The probability of more than 6 occurrences is approximately \( 0.1954 \).
- **Part (c)**: The probability of 3 or fewer occurrences is approximately \( 0.3097 \).

Would you like further details or explanations on any of the calculations?

Here are some related questions to explore:
1. How does changing \( \lambda \) affect the probabilities in a Poisson distribution?
2. Can you calculate the probability of exactly 7 occurrences using the same method?
3. What is the cumulative probability of 5 or fewer occurrences?
4. How can the Poisson distribution be used to model real-world scenarios?
5. What is the difference between a Poisson distribution and a binomial distribution?

**Tip**: The Poisson distribution is often used to model rare events over a fixed interval of time or space.

Consider a Poisson probability distribution with λ = 4.7. Determine the following probabilities: a) exactly 5 occurrences, b) more than 6 occurrences, c) 3 or fewer occurrences.

Learn how to solve a Poisson probability distribution problem with λ = 4.7. This problem involves calculating the probabilities of exactly 5 occurrences, more than 6 occurrences, and 3 or fewer occurrences using the Poisson formula. Suitable for college-level students.

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