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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.3.18c
Chapter 4, Problem 4.3.18c

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.


Living Donor Transplants The mean number of organ transplants from living donors performed per day in the United States in 2020 was about 16. Find the probability that the number of organ transplants from living donors performed on any given day is (c) no more than 10. (Source: Organ Procurement and Transplantation Network)

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Step 1: Identify the type of distribution to use. Since the problem involves the mean number of events (organ transplants) occurring per day and the events are independent, the Poisson distribution is appropriate. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space when the mean rate of occurrence is known.
Step 2: Write down the formula for the Poisson probability. The probability of observing exactly k events in a Poisson distribution is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).
Step 3: Define the problem in terms of the Poisson distribution. Here, λ = 16 (the mean number of organ transplants per day), and we are tasked with finding the probability that the number of transplants is no more than 10. This means we need to calculate P(X ≤ 10), which is the cumulative probability for k = 0, 1, 2, ..., 10.
Step 4: Use the cumulative probability formula. To find P(X ≤ 10), sum the probabilities for each value of k from 0 to 10: P(X ≤ 10) = P(X = 0) + P(X = 1) + ... + P(X = 10). For each term, substitute the values of λ and k into the Poisson formula: P(X = k) = (16^k * e^(-16)) / k!.
Step 5: Use technology or a Poisson distribution table to compute the cumulative probability. Alternatively, use statistical software or a calculator with a Poisson cumulative distribution function (CDF) to directly compute P(X ≤ 10). Once the probability is found, compare it to a threshold (e.g., 0.05) to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05.

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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 organ transplants per day. The key parameter is the mean (λ), which represents the average number of occurrences in the interval.
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Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring, expressed as a number between 0 and 1. In the context of the Poisson distribution, this is done using the formula P(X=k) = (e^(-λ) * λ^k) / k!, where k is the number of occurrences, λ is the mean, and e is Euler's number. Understanding how to apply this formula is essential for finding the required probabilities.
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Unusual Events

An event is considered unusual if its probability is significantly low, often defined as less than 5%. In statistical analysis, determining whether an event is unusual helps in understanding its significance in the context of the data. For the given problem, after calculating the probability of 10 or fewer transplants, one would assess if this probability falls below the threshold to classify it as unusual.
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Related Practice
Textbook Question

Manufacturing An assembly line produces 10,000 automobile parts. Twenty percent of the parts are defective. An inspector randomly selects 10 of the parts


b. Because the sample is only 0.1% of the population, treat the events as independent and use the binomial probability formula to approximate the probability that none of the selected parts are defective.

Textbook Question

Unusual Events In Exercises 37 and 38, find the indicated probabilities. Then determine if the event is unusual. Explain your reasoning.


Rock-Paper-Scissors The probability of winning a game of rock-paper-scissors is 1/3. You play nine games of rock-paper-scissors. Find the probability that the number of games you win is (c) less than two.

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Textbook Question

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.


Living Donor Transplants The mean number of organ transplants from living donors performed per day in the United States in 2020 was about 16. Find the probability that the number of organ transplants from living donors performed on any given day is (b) at least eight (Source: Organ Procurement and Transplantation Network)

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Textbook Question

Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and failures, the probability of selecting a sample of size that has successes and failures is given by

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In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Use the above formula to find the probability that (c) two microchips are defective and one is not defective.

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Textbook Question

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (c) from one to three HD televisions,

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Textbook Question

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.


Oil Tankers In the month of June 2021, 240 oil tankers stop at a port city. No oil tanker visits more than once. Find the probability that the number of oil tankers that stop on any given day in June is (b) at most three

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