Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.1.21a

Chapter 4, Problem 4.1.21a

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

Verified step by step guidance

Step 1: Recall the probability distribution created in Exercise 19. A probability distribution lists all possible outcomes (e.g., the number of HD televisions in a household) and their corresponding probabilities. Ensure that the probabilities sum to 1.

Step 2: Identify the outcomes of interest. In this case, we are looking for households with either one or two HD televisions. These correspond to the outcomes '1' and '2' in the probability distribution.

Step 3: Locate the probabilities associated with the outcomes '1' and '2' in the probability distribution. These probabilities are denoted as P(X=1) and P(X=2), where X represents the number of HD televisions.

Step 4: Add the probabilities of the outcomes '1' and '2' together to find the total probability. Use the formula: P(X=1 or X=2) = P(X=1) + P(X=2).

Step 5: Verify that the result is valid by ensuring it falls within the range of 0 to 1, as probabilities cannot exceed these bounds.

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Key Concepts

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

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It can be discrete, where outcomes are distinct and countable, or continuous, where outcomes can take any value within a range. Understanding how to construct and interpret a probability distribution is essential for calculating probabilities related to specific events.

Random Selection

Random selection refers to the process of choosing individuals or items from a population in such a way that each member has an equal chance of being selected. This concept is crucial in statistics as it helps ensure that the sample is representative of the population, allowing for valid inferences and conclusions. In the context of the question, it implies that each household has an equal opportunity to be chosen when assessing the number of HD televisions.

Event Probability

Event probability is the measure of the likelihood that a specific event will occur, expressed as a number between 0 and 1. In this case, it involves calculating the probability of selecting a household with one or two HD televisions based on the provided distribution. Understanding how to compute event probabilities is fundamental for making predictions and informed decisions based on statistical data.

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Related Practice

Textbook Question

Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

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.)

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 increases to five arrivals per minute, but the store can still process only four per minute. Generate a list of 20 random numbers with a Poisson distribution for mu = 5 . Then create a table that shows the number of customers waiting at the end of 20 minutes.

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

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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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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