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Ch. 3 - Probability
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. 3 - ProbabilityProblem 3.2.25b
Chapter 3, Problem 3.2.25b

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
25. Best President In a sample of 1500 adult U.S. citizens, 270 said that Barack Obama was the best president in U.S. history. Two adult U.S. citizens are selected at random.
(Adapted from YouGov)
b. Find the probability that neither adult U.S. citizen says that Barack Obama was the best president in U.S. history."

Verified step by step guidance
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Step 1: Understand the problem. We are tasked with finding the probability that neither of the two randomly selected adult U.S. citizens believes Barack Obama was the best president in U.S. history. This involves using the Multiplication Rule for probabilities.
Step 2: Calculate the probability that a single adult U.S. citizen does not believe Barack Obama was the best president. From the sample, 270 out of 1500 said he was the best president. Therefore, the probability that a single person does not believe this is given by \( P(\text{not Obama}) = 1 - \frac{270}{1500} \).
Step 3: Recognize that the two selections are made without replacement. This means the probability for the second selection depends on the outcome of the first selection. For the first selection, the probability that the person does not believe Barack Obama was the best president is \( P(\text{not Obama}) \).
Step 4: For the second selection, if the first person selected does not believe Barack Obama was the best president, the total number of people who do not believe this decreases by 1, and the total population decreases by 1. Thus, the probability for the second selection is \( P(\text{not Obama on second draw}) = \frac{N_{\text{not Obama}} - 1}{1500 - 1} \), where \( N_{\text{not Obama}} \) is the number of people who do not believe Barack Obama was the best president.
Step 5: Apply the Multiplication Rule. The probability that neither of the two selected individuals believes Barack Obama was the best president is the product of the probabilities for the first and second selections: \( P(\text{neither}) = P(\text{not Obama on first draw}) \times P(\text{not Obama on second draw}) \). Substitute the values from the previous steps to compute the final probability.

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Key Concepts

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

Multiplication Rule

The Multiplication Rule in probability states that the probability of two independent events both occurring is the product of their individual probabilities. In this context, it helps calculate the likelihood of multiple selections from a population, where the outcome of one selection does not affect the other.
Recommended video:
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Multiplication Rule: Dependent Events

Probability of an Event

Probability quantifies the likelihood of an event occurring, expressed as a number between 0 and 1. In this scenario, we need to determine the probability that a randomly selected adult does not believe Barack Obama was the best president, which involves calculating the complement of the probability that they do.
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05:54
Probability of Multiple Independent Events

Complement Rule

The Complement Rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring. For this question, if we know the probability of an adult saying Obama is the best president, we can easily find the probability that neither of the two selected adults holds that opinion by applying this rule.
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Conditional Probability Rule
Related Practice
Textbook Question

25. Working from Home The table shows the results of a survey that asked 1811 people how often they work from home. A person is selected at random from the sample. Find the probability of each event.

b. The person is female or does not work from home.

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Textbook Question

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

27. Blood Types The probability that a person of Asian descent in the United States has type O+ blood is 39%. At random, six people of Asian descent in the United States are selected. (Source: American National Red Cross)

b. Find the probability that none of the six have type O+ blood."

Textbook Question

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

24. Knowing a Person Who Was Murdered In a sample of 11,771 children ages 2 to 17, 8% have lost a friend or relative to murder. Four children are selected at random. (Adapted from University of New Hampshire)

b. Find the probability that none of the four has lost a friend or relative to murder."

Textbook Question

Politics The responses of 1500 U.S. adults to a survey that asked them to state their own political viewpoints are shown in the Pareto chart. Find the probability of each event.(Adapted from YouGov)

b. Randomly selecting a person from the sample who is conservative or very conservative

Textbook Question

88. Individual Stock Price An individual stock is selected at random from the portfolio represented by the box-and-whisker plot shown. Find the probability that the stock price is between \$21 and \$50.

Textbook Question

Officers The offices of president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 14 candidates. Six of the candidates are members of the debate team.

b. What is the probability that none of the offices are filled by members of the debate team?