To solve this problem, we will use the Poisson distribution formula:

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

Where:

• $\lambda$ is the mean number of occurrences (here, $\lambda = 7.2$).
• $k$ is the number of occurrences (in part (a), $k = 5$).
• $e$ is the base of the natural logarithm, approximately 2.71828.

Part a: Find the probability that, in a year, there will be 5 hurricanes.

The probability is calculated using the Poisson formula. We are given:

• $\lambda = 7.2$
• $k = 5$

Now, applying the values to the formula:

$P(X = 5) = \frac{{7.2^5 e^{-7.2}}}{5!}$

Let me calculate this probability.The probability that there will be 5 hurricanes in a year is approximately 0.120 (rounded to three decimal places).

Part b: In a 55-year period, how many years are expected to have 5 hurricanes?

We can calculate this by multiplying the probability found in part (a) by the total number of years (55 years):

$\text{Expected years with 5 hurricanes} = P(X = 5) \times 55$

Let me calculate that.In a 55-year period, we would expect approximately 6.62 years to have 5 hurricanes.

Part c: Comparison to the actual period

In the actual 55-year period, there were 6 years with 5 hurricanes. The expected number of years with 5 hurricanes, based on the Poisson distribution, was 6.62 years.

The observed value (6 years) is very close to the expected value (6.62 years). Therefore, it seems that the Poisson distribution works quite well in this case.

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Do you want more details or have any other questions?

Here are 5 related questions to explore:

1. How is the Poisson distribution applied in other real-world phenomena?
2. What is the probability of having exactly 7 hurricanes in a year?
3. How would the probability change if the mean number of hurricanes were 10 instead of 7.2?
4. How can we calculate the probability of having fewer than 3 hurricanes in a year?
5. What are the assumptions behind the Poisson distribution, and when is it appropriate to use it?

Tip: The Poisson distribution is ideal for modeling rare events that occur independently over a fixed period of time, such as natural disasters or system failures.