Question

Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 7.5 per year.

A) Find the probability that, in a year, there will be 6 hurricanes.

B) In a 45-year period, how many years are expected to have 6 hurricanes?

C) How does the result from part (B) compare to a recent period of 45 years in which 6 years had 6 hurricanes? Does the Poisson distribution work well here?

Answer 1

The probability of there being 6 hurricanes in a year is approximately 0.09735. In a 45-year period, we would expect 7.5 years to have 6 hurricanes. The Poisson distribution works well in this case.

Step-by-step explanation:

To find the probability of there being 6 hurricanes in a year, we can use the Poisson probability formula. The formula is:

P(x;μ) = (e^(-μ) * μ^x) / x!

Where x is the number of events, μ is the mean number of events, e is Euler's number, and x! represents factorial of x. Substituting the values into the formula, we get:

P(x=6;μ=7.5) = (e^(-7.5) * 7.5^6) / 6!

Calculating this expression, we find that the probability of there being 6 hurricanes in a year is approximately 0.09735.

To find the expected number of years with 6 hurricanes in a 45-year period, we can use the concept of expected value. The expected value is equal to the mean of the distribution, which in this case is 7.5. So, in a 45-year period, we would expect 7.5 years to have 6 hurricanes.

In a recent 45-year period where 6 years had 6 hurricanes, we can compare this to the expected value of 7.5 years. While the observed number of years (6) is slightly lower than the expected value, it is within the realm of what can be expected due to the random nature of the Poisson distribution. Therefore, the Poisson distribution works well in this case.