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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 6a
Chapter 5, Problem 6a

 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 no hurricanes.

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Step 1: Recall the formula for the Poisson probability distribution: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of occurrences, k is the number of occurrences we are interested in, and e is the base of the natural logarithm (approximately 2.718).
Step 2: Identify the given values from the problem. Here, λ = 5.5 (the mean number of hurricanes per year) and k = 0 (since we are finding the probability of no hurricanes in a year).
Step 3: Substitute the values into the formula. This gives P(X = 0) = (5.5^0 * e^(-5.5)) / 0!. Note that any number raised to the power of 0 is 1, and 0! (zero factorial) is also equal to 1.
Step 4: Simplify the expression. The numerator becomes 1 (since 5.5^0 = 1), and the denominator is 1 (since 0! = 1). This leaves P(X = 0) = e^(-5.5).
Step 5: To find the final probability, calculate e^(-5.5) using a calculator or software. This will give the probability of having no hurricanes in a year.

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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 of each other.
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Intro to Frequency Distributions

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the 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.
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Calculating Probability with Poisson

To find the probability of observing a specific number of events in a Poisson distribution, the formula P(X=k) = (e^(-λ) * λ^k) / k! is used, where P(X=k) is the probability of k events occurring, e is the base of the natural logarithm, and k! is the factorial of k. For this question, to find the probability of zero hurricanes, k would be 0.
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Related Practice
Textbook Question

In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).



Probability Find the probability that at least one of the subjects is a sleepwalker.

Textbook Question

 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


b. In a 118-year period, how many years are expected to have no hurricanes?

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Textbook Question

Texting and Driving. In Exercises 21–26, refer to the accompanying table, which describes probabilities for groups of five drivers. The random variable x is the number of drivers in a group who say that they text while driving (based on data from an Arity survey of drivers).

For groups of five drivers, find the mean and standard deviation for the numbers of drivers who say that they text while driving.

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Textbook Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.


e. What do the results suggest about how the clerk met the requirement of using a random method to assign the order of candidates’ names on voting ballots?

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Textbook Question

Texting and Driving. In Exercises 21–26, refer to the accompanying table, which describes probabilities for groups of five drivers. The random variable x is the number of drivers in a group who say that they text while driving (based on data from an Arity survey of drivers).

Range Rule of Thumb for Significant Events

Use the range rule of thumb to determine whether 1 is a significantly low number of drivers who say that they text while driving.

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Textbook Question

In Exercises 6–10, refer to the accompanying table, which describes the numbers of adults in groups of five who reported sleepwalking (based on data from “Prevalence and Comorbidity of Nocturnal Wandering In the U.S. Adult General Population,” by Ohayon et al., Neurology, Vol. 78, No. 20).


Does the table describe a probability distribution?