Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 7.2  per year. 
a. Find the probability​ that, in a​ year, there will be  5 hurricanes.
b. In a  55​-year ​period, how many years are expected to have 55 ​hurricanes?
c. How does the result from part​ (b) compare to a recent period of  55 years in which  6 years had  5 ​hurricanes? Does the Poisson distribution work well​ here?
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Part 1
a. The probability is enter your response here.
​(Round to three decimal places as​ needed.)

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.

Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 7.2 per year.
a. Find the probability​ that, in a​ year, there will be 5 hurricanes.
b. In a 55​-year ​period, how many years are expected to have 5 ​hurricanes?
c. How does the result from part​ (b) compare to a recent period of 55 years in which 6 years had 5 ​hurricanes? Does the Poisson distribution work well​ here?

Solve a real-world problem using the Poisson distribution to calculate the probability of having 5 hurricanes in a year when the mean is 7.2. Find out how many years in a 55-year period are expected to have 5 hurricanes and compare it to actual data. Suitable for Grades 10-12.

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