"Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Hurricanes The mean number of hurricanes to strike the U.S. mainland per year from 1851 through 2020 was about 1.8. Find the probability that the number of hurricanes striking the U.S. mainland in any given year from 1851 through 2020 is (a) exactly one, (b) at most one, and (c) more than one. (Source: National Oceanic & Atmospheric Administration)"

Verified step by step guidance

Step 1: Identify the type of probability distribution to use. Since the problem involves the mean number of hurricanes per year (1.8) and asks for probabilities of discrete events (exactly one, at most one, more than one), the Poisson distribution is appropriate. The Poisson distribution is used to model the number of occurrences of an event in a fixed interval of time or space, given a known average rate (λ).

Step 2: Write the formula for the Poisson probability distribution. The probability of observing k events in a Poisson distribution is given by:

P(k) = (λ

^k * e^-λ) / k!, where λ is the mean number of occurrences, k is the number of occurrences, and e is the base of the natural logarithm (approximately 2.718).

Step 3: Solve part (a) for exactly one hurricane. Substitute λ = 1.8 and k = 1 into the formula:

P(1) = (1.8

¹ * e^-1.8) / 1!. Simplify the expression to find the probability.

Step 4: Solve part (b) for at most one hurricane. 'At most one' means k = 0 or k = 1. Calculate the probabilities for k = 0 and k = 1 using the formula, then add them together:

P(k ≤ 1) = P(0) + P(1)

. For k = 0, substitute λ = 1.8 and k = 0 into the formula:

P(0) = (1.8

⁰ * e^-1.8) / 0!. For k = 1, use the result from part (a).

Step 5: Solve part (c) for more than one hurricane. 'More than one' means k > 1. Use the complement rule:

P(k > 1) = 1 - P(k ≤ 1)

. Use the result from part (b) to calculate the complement probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 the number of hurricanes in a year, where the events are independent. 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 and Unusual Events

The mean, or expected value, of a probability distribution is a measure of the central tendency, representing the average outcome over a long period. In the context of the Poisson distribution, the mean also indicates the average number of events (e.g., hurricanes) expected in a given timeframe. An event is considered unusual if its probability is significantly low, often defined as less than 5%, prompting further investigation into its occurrence.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. In the context of the question, calculating the cumulative probability for hurricanes striking the U.S. mainland involves finding the probability of having at most one hurricane in a year. This is done by summing the probabilities of having zero and one hurricane, which can be efficiently calculated using the Poisson distribution.

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