In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 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 7 hurricanes.

Verified step by step guidance

1. Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence (mean). The formula for the Poisson probability is P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of occurrences, k is the number of occurrences, and e is the mathematical constant approximately equal to 2.718.
2. Step 2: Identify the given values. From the problem, the mean number of hurricanes per year (λ) is 5.5, and we are tasked with finding the probability of exactly 7 hurricanes (k = 7).
3. Step 3: Substitute the values into the Poisson formula. Replace λ with 5.5 and k with 7 in the formula: P(X = 7) = (5.5^7 * e^(-5.5)) / 7!.
4. Step 4: Break down the calculation. Compute each component step-by-step: (1) Calculate 5.5^7, (2) Compute e^(-5.5), (3) Find the factorial of 7 (7!), and (4) Divide the product of 5.5^7 and e^(-5.5) by 7!.
5. Step 5: Interpret the result. Once the probability is calculated, it represents the likelihood of observing exactly 7 hurricanes in a year under the given conditions. This value can be compared to other probabilities to understand the distribution of hurricane occurrences.

Key Concepts

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as natural disasters, where the events occur independently. The formula for the Poisson probability mass function is P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of occurrences, and e is Euler's number.

Mean (λ)

In the context of the Poisson distribution, the mean (denoted as λ) represents the average number of occurrences of the event in a specified interval. For this question, λ is given as 5.5, indicating that, on average, there are 5.5 hurricanes per year in the United States. This parameter is crucial for calculating the probabilities of different outcomes using the Poisson formula.

Probability Calculation

To find the probability of a specific number of events occurring in a Poisson distribution, one uses the Poisson probability mass function. For this question, to find the probability of exactly 7 hurricanes in a year, you would substitute k=7 and λ=5.5 into the formula. This calculation will yield the likelihood of observing that exact number of hurricanes, which is essential for understanding the variability and risk associated with hurricane occurrences.