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Part 1
Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 5.6 per year. 
a. Find the probability​ that, in a​ year, there will be 3 hurricanes.

Given that the number of hurricanes follows a Poisson distribution with a mean (\(\lambda\)) of 5.6 hurricanes per year, we are asked to find the probability that there will be 3 hurricanes in a given year. 

The Poisson probability mass function (PMF) is given by:

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

where:
- \(\lambda = 5.6\) (the mean number of hurricanes per year),
- \(k = 3\) (the desired number of hurricanes),
- \(e\) is Euler's number (approximately 2.71828).

### Step-by-step Solution:

1. Substitute the values \(\lambda = 5.6\) and \(k = 3\) into the Poisson formula:

\[
P(X = 3) = \frac{5.6^3 e^{-5.6}}{3!}
\]

2. Calculate \(5.6^3\):

\[
5.6^3 = 175.616
\]

3. Compute \(e^{-5.6}\):

\[
e^{-5.6} \approx 0.003698
\]

4. Find \(3!\) (factorial of 3):

\[
3! = 3 \times 2 \times 1 = 6
\]

5. Now, plug everything into the formula:

\[
P(X = 3) = \frac{175.616 \times 0.003698}{6} \approx \frac{0.6491}{6} \approx 0.1082
\]

### Final Answer:
The probability that there will be exactly 3 hurricanes in a year is approximately **0.1082**.

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Would you like further details on this, or have any additional questions?

Here are 5 related questions:
1. How would the probability change if we wanted to know the chance of 4 hurricanes?
2. What is the cumulative probability of having fewer than 3 hurricanes in a year?
3. How can we use the Poisson distribution to model events other than hurricanes?
4. If the mean number of hurricanes changed to 6 per year, how would that affect the probability of having exactly 3 hurricanes?
5. How would you use the Poisson distribution to calculate the probability of no hurricanes in a year?

**Tip:** When calculating probabilities with Poisson distribution, the mean \(\lambda\) plays a key role in determining how spread out the possible values are around this mean.

Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 5.6 per year. a. Find the probability​ that, in a​ year, there will be 3 hurricanes.

Calculate the probability of exactly 3 hurricanes in a year using the Poisson distribution, where the mean number of hurricanes is 5.6. This step-by-step solution covers key probability concepts, including the Poisson probability mass function, and is ideal for Grades 11-12.

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