Let's break down the problem and solve it step by step using the Poisson distribution.

Given:

• The mean number of hurricanes per year: $\lambda = 6.4$.
• The Poisson distribution formula is:

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

Where:

• $X$ is the number of occurrences (hurricanes in this case).
• $k$ is the specific number of hurricanes you are calculating the probability for (in this case $k = 5$).
• $\lambda$ is the average number of occurrences (here, $\lambda = 6.4$).
• $e$ is Euler's number, approximately 2.71828.

Part a. Finding the probability of 5 hurricanes in a year:

We need to calculate $P(X = 5)$ when $\lambda = 6.4$.

$P(X = 5) = \frac{6.4^5 e^{-6.4}}{5!}$

Let's calculate this.

Part b. Expected number of years with 5 hurricanes in a 55-year period:

The expected number of years can be found by multiplying the probability from part a by the number of years:

$E = 55 \times P(X = 5)$

Part c. Comparison with the actual data (8 years had 5 hurricanes):

Here, you're asked to compare the expected number of years (from part b) with the actual value of 8 years. Based on the options provided (close to or very different from), you will determine if the Poisson distribution is a good fit for this data.

I'll now calculate the values for parts a and b.### Part a: The probability that there will be exactly 5 hurricanes in a year is approximately 0.149 (rounded to three decimal places).

Part b:

The expected number of years with 5 hurricanes in a 55-year period is approximately 8.2 (rounded to one decimal place).

Part c:

Since the expected number of years with 5 hurricanes is 8.2, which is very close to the actual value of 8 years, you would select:

• (1) close to
• (2) does (as the Poisson distribution appears to work well in this situation).

Would you like to explore more details about this? Here are some related questions to dive deeper:

1. How does the Poisson distribution differ from the binomial distribution in modeling rare events?
2. What assumptions are necessary for the Poisson distribution to apply?
3. How would the probability change if the mean number of hurricanes were higher?
4. Can the Poisson distribution model be applied for more extreme natural events (e.g., earthquakes)?
5. What happens to the Poisson distribution if the average rate increases significantly?

Tip: When working with real-world data, comparing predicted probabilities to actual outcomes helps validate whether the chosen model (like Poisson distribution) is a good fit.