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.