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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 4.3.28a
Chapter 4, Problem 4.3.28a

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 (a) all three microchips are not defective

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Step 1: Understand the problem. We are tasked with finding the probability that all three microchips selected are not defective using the hypergeometric distribution formula. The formula is P(x) = ((kCx) * ((N-k)C(n-x))) / (NCn), where N is the total population size, k is the number of successes in the population, n is the sample size, and x is the number of successes in the sample.
Step 2: Identify the values from the problem. Here, N = 15 (total microchips), k = 13 (non-defective microchips, considered as successes), n = 3 (sample size), and x = 3 (all three microchips are not defective).
Step 3: Compute the numerator of the formula. This involves two combinations: (kCx) and ((N-k)C(n-x)). For (kCx), calculate the number of ways to choose x successes from k successes. For ((N-k)C(n-x)), calculate the number of ways to choose (n-x) failures from (N-k) failures.
Step 4: Compute the denominator of the formula. This is the total number of ways to choose n items from N items, represented as NCn.
Step 5: Substitute the computed values into the formula and simplify. The probability P(x) is obtained by dividing the numerator by the denominator. This will give the probability that all three microchips selected are not defective.

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Key Concepts

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

Hypergeometric Distribution

The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population without replacement. It is characterized by the population size (N), the number of successes in the population (k), the sample size (n), and the number of observed successes (x). Unlike the binomial distribution, where sampling is done with replacement, the hypergeometric distribution accounts for the changing probabilities as items are drawn.
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Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In the context of the hypergeometric distribution, combinatorial notation (nCr) is used to calculate the number of ways to choose r successes from a total of n items. This is crucial for determining the probabilities of different outcomes when sampling from a finite population.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring. In the hypergeometric distribution, this is done using the formula provided, which combines the number of ways to choose successes and failures from the population. Understanding how to apply this formula is essential for solving problems related to sampling without replacement, such as finding the probability of selecting non-defective microchips from a shipment.
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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


a. Use the Multiplication Rule (discussed in Section 3.2) to find the probability that none of the selected parts are defective. (Note that the events are dependent.)

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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 (a) exactly eight

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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 (a) exactly 12 (Source: Organ Procurement and Transplantation Network)

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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 (a) exactly five

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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 (a) one or two HD televisions

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Textbook Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of customers who arrive at the checkout counters each minute is 4. Create a Poisson distribution with mu = 4 for x = 0 to 20. Compare your results with the histogram shown at the upper right.

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