To solve the given problem using the Poisson distribution, let's break it down.

Key Information:

• The mean number of hurricanes per year ($\lambda$) is 6.3.
• We are using the Poisson distribution, where the probability of observing $k$ events is given by:

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

where:

• $k$ is the number of events (hurricanes in this case),
• $\lambda$ is the average rate (mean),
• $e$ is Euler's number ($\approx 2.71828$).

(a) Find the probability that, in a year, there will be 4 hurricanes.

We need to calculate $P(4; 6.3)$, i.e., the probability that $k = 4$ hurricanes occur when the mean number of hurricanes per year is $\lambda = 6.3$.

$P(4; 6.3) = \frac{6.3^4 e^{-6.3}}{4!}$

First, calculate the components:

• $6.3^4 = 1575.0621$,
• $e^{-6.3} \approx 0.001834$,
• $4! = 24$.

Now plug these into the formula:

$P(4; 6.3) = \frac{1575.0621 \times 0.001834}{24} = \frac{2.8881}{24} \approx 0.120.$

So, the probability is approximately 0.120.

(b) In a 55-year period, how many years are expected to have 4 hurricanes?

We calculate the expected number of years with 4 hurricanes in a 55-year period by multiplying the probability from part (a) by 55:

$\text{Expected years} = 55 \times 0.120 = 6.6 \text{ years}.$

(c) Compare the result from (b) to a recent period of 55 years in which 6 years had 4 hurricanes. Does the Poisson distribution work well here?

The expected number of years with 4 hurricanes is 6.6, while the actual number of years with 4 hurricanes is 6. The Poisson distribution seems to work well in this case, as the expected value is very close to the observed value.

Would you like further details on any part? Here are five related questions:

1. How do you calculate factorials in the Poisson distribution?
2. What happens to the probability as the number of hurricanes $k$ increases?
3. Can the Poisson distribution be used for very large values of $k$?
4. How does changing the mean ($\lambda$) affect the probability?
5. What are some real-world limitations of the Poisson model for hurricane predictions?

Tip: For Poisson distributions, the variance is equal to the mean, which can be useful in assessing the spread of data.