Question

Hurricanes. In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example 1; and proceed to find the indicated probability.

Hurricanes

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 the recent period of 55 years in which 8 years had 5 hurricanes? Does the Poisson distribution work well here?

Short answer

a.The probability of 5 hurricanes in a year is equal to 0.158.

b. Thus, the expected number of years to have 5 hurricanes in a 55-year period is equal to 8.7 years.

c. The expected number of years that have 5 hurricanes is approximately equal to the actual number of years that have 5 hurricanes in a 55-year period.Since the expected and the actual values are approximately equal, the number of hurricanes is well-modeled by the Poisson distribution.

Step-by-step solution

Given information

The mean number of Atlantic hurricanes in the United States is given to be equal to 6.1 per year.

Poisson probability

a.

Let X be the number of Atlantic hurricanes in one year. Here, X follows a Poisson distribution with mean equal to\({\kern 1pt} \mu = 6.1\).

The probability of 5 hurricanes in a year is computed below:

\[\begin{aligned}{c}P\left( x \right) = \frac{{{\mu ^x}{e^{ - \mu }}}}{{x!}}\\P\left( 5 \right) = \frac{{{{\left( {6.1} \right)}^5}{{\left( {2.71828} \right)}^{ - 6.1}}}}{{5!}}\\ = 0.15786\\ \approx 0.158\end{aligned}\]

Therefore, the probability of 5 hurricanes in a year is equal to 0.158.

Expected number of hurricanes

b.

The expected number of years to have 5 hurricanes in a 55-year period is computed below:

\(\begin{aligned}{c}55 \times P\left( 5 \right) = 55 \times 0.158\\ = 8.69\\ \approx 8.7\end{aligned}\)

Thus, the expected number of years to have 5 hurricanes in a 55-year period is equal to 8.7 years.

Step 4:Comparison of actual and expected values

c.

It is given that the actual number of years that had 5 hurricanes in the recent 55-year period is equal to 8.

The expected number of years that have 5 hurricanes in a 55-year period is equal to 8.7.

Thus, the expected number of years is equal to the actual number of years.

Since the expected and the actual values are approximately equal, the number of hurricanes is well-modeled by the Poisson distribution.