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 4.1.21d

Chapter 4, Problem 4.1.21d

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (d) at most 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, as this is a requirement for a valid probability distribution.

Step 2: Identify the outcomes that correspond to 'at most two HD televisions.' This means you are interested in the probabilities for households with 0, 1, or 2 HD televisions.

Step 3: Write the mathematical expression for the probability of 'at most two HD televisions' as the sum of the probabilities for these outcomes: \( P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) \).

Step 4: Substitute the probabilities for \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \) from the probability distribution into the equation. Ensure that you use the correct values from the distribution.

Step 5: Add the probabilities together to compute the total probability of selecting a household with at most two HD televisions. Verify that the result is a valid probability (i.e., it lies between 0 and 1).

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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.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. In the context of the question, finding the probability of selecting a household with at most two HD televisions involves calculating the cumulative probability for the values 0, 1, and 2. This concept is crucial for determining the likelihood of outcomes within a defined range.

Random Selection

Random selection is a fundamental principle in statistics that ensures each member of a population has an equal chance of being chosen. This concept is vital for ensuring that the results of probability calculations are valid and representative of the entire population. In the context of the question, it implies that the households are selected without bias, allowing for accurate probability assessments.

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Textbook Question

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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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