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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 5.3.8b
Chapter 5, Problem 5.3.8b

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.
b. In a 118-year period, how many years are expected to have 10 hurricanes?

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1
Understand the problem: The Poisson distribution is used to model the number of events (hurricanes) occurring in a fixed interval of time (years). The mean number of hurricanes per year is given as \( \lambda = 5.5 \). We are tasked with finding the expected number of years with 10 hurricanes over a 118-year period.
Recall the formula for the Poisson probability mass function (PMF): \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \), where \( k \) is the number of events (hurricanes), \( \lambda \) is the mean number of events, and \( e \) is the base of the natural logarithm.
Calculate the probability of exactly 10 hurricanes in a single year using the Poisson PMF formula. Substitute \( \lambda = 5.5 \) and \( k = 10 \) into the formula: \( P(X = 10) = \frac{e^{-5.5} (5.5)^{10}}{10!} \). Simplify the expression but do not compute the final value.
To find the expected number of years with 10 hurricanes over a 118-year period, multiply the probability of 10 hurricanes in a single year by the total number of years: \( \text{Expected years} = P(X = 10) \times 118 \).
Substitute the previously calculated \( P(X = 10) \) into the formula \( \text{Expected years} = P(X = 10) \times 118 \). Simplify the expression but do not compute the final value.

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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

Expected Value

The expected value is a key concept in probability that represents the average outcome of a random variable over a large number of trials. In the context of the Poisson distribution, the expected value is equal to the mean (λ), which indicates the average number of occurrences—in this case, the average number of hurricanes per year.
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Expected Value (Mean) of Random Variables

Rate of Occurrence

The rate of occurrence in a Poisson distribution refers to the average number of events (hurricanes) expected to happen in a specified time frame (one year). In this scenario, with a mean of 5.5 hurricanes per year, this rate helps determine the likelihood of observing a specific number of hurricanes over a longer period, such as 118 years.
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Related Practice
Textbook Question

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.


Gender Selection Assume that the groups consist of 36 couples.


b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

Textbook Question

Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.


Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.


b. Beginning with WWC, make a complete list of the different possible arrangements of two wrong answers and one correct answer, and then find the probability for each entry in the list.

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Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.


Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.


b. Find the probability that in a single day, there are 16 births.

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 7 hurricanes?

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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.

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

Textbook Question

In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).


Hybrids Assume that offspring peas are randomly selected in groups of 16.


b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

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