Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

Statistics

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

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1:43 minutes

Problem 4.1.21d

Textbook Question

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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Related Practice

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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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

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (b) two or more HD televisions

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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

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 arrivals per minute is four. Find the probability that

a. three, four, or five customers will arrive during the third minute.

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Textbook Question

An online magazine finds that the mean number of typographical errors per page is five. Find the probability that the number of typographical errors found on any given page is (a) exactly five, (b) less than five, and (c) exactly zero.

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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 arrivals per minute is four. Find the probability that

c. one customer is waiting in line after one minute and no customers are waiting in line after the second minute..

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Textbook Question

Testing a Drug A drug manufacturer claims that a drug cures a rare skin disease 75% of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted. Use this information in Exercises 31 and 32.

Find the probability that the claim will be accepted, assuming that the actual probability that the drug cures the skin disease is 65%.

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