Question

Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 6.9 per year. a. Find the probability​ that, in a​ year, there will be 4 hurricanes. b. In a 45​-year ​period, how many years are expected to have 4 ​hurricanes? c. How does the result from part​ (b) compare to a recent period of 45 years in which 4 years had 4 ​hurricanes? Does the Poisson distribution work well​ here? a. The probability is nothing. ​(Round to three decimal places as​ needed.)

Answer 1

a) 9.52% probability​ that, in a​ year, there will be 4 hurricanes.

b) 4.284 years are expected to have 4 ​hurricanes.

c) The value of 4 is very close to the expected value of 4.284, so the Poisson distribution works well here.

Explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\mu` is the mean in the given time interval.

6.9 per year.

This means that
`\mu = 6.9`

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

This is P(X = 4).

So

`P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)`

`P(X = 4) = (e^(-6.9)*(6.9)^(4))/((4)!)`

`P(X = 4) = 0.0952`

9.52% probability​ that, in a​ year, there will be 4 hurricanes.

b. In a 45​-year ​period, how many years are expected to have 4 ​hurricanes?

For each year, the probability is 0.0952.

Multiplying by 45

45*0.0952 = 4.284.

4.284 years are expected to have 4 ​hurricanes.

c. How does the result from part​ (b) compare to a recent period of 45 years in which 4 years had 4 ​hurricanes? Does the Poisson distribution work well​ here?

The value of 4 is very close to the expected value of 4.284, so the Poisson distribution works well here.