Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 3 - Probability

Problem 3.2.24b

Chapter 3, Problem 3.2.24b

"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."

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that none of the four randomly selected children has lost a friend or relative to murder. This involves using the Multiplication Rule for independent events.

Step 2: Define the probability of the complementary event. The problem states that 8% of children have lost a friend or relative to murder. Therefore, the probability that a child has NOT lost a friend or relative to murder is 1 - 0.08 = 0.92.

Step 3: Recognize that the four children are selected independently. This means the outcome for one child does not affect the outcomes for the others. The Multiplication Rule for independent events states that the probability of all events occurring is the product of their individual probabilities.

Step 4: Apply the Multiplication Rule. To find the probability that none of the four children has lost a friend or relative to murder, multiply the probability of the complementary event (0.92) by itself four times: $ P(\text{none}) = 0.92 \times 0.92 \times 0.92 \times 0.92 $.

Step 5: Simplify the expression. This can also be written as $ P(\text{none}) = 0.92^4 $. Use this formula to calculate the final probability if needed.

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Key Concepts

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

Multiplication Rule of Probability

The Multiplication Rule 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 children not having lost a friend or relative to murder by multiplying the probabilities of each child independently not experiencing this loss.

Complementary Probability

Complementary probability refers to the probability of an event not occurring. In this scenario, if 8% of children have lost someone to murder, then 92% have not. This concept is crucial for determining the probability that none of the selected children have experienced this loss, as it allows us to use the probability of the complementary event.

Independent Events

Independent events are those whose outcomes do not affect each other. In this problem, the selection of each child is independent, meaning the probability of one child not having lost someone to murder does not change based on the selections of the others. This independence is essential for applying the Multiplication Rule correctly.

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Probability of Multiple Independent Events

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.

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."

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.

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)

c. Find the probability that at least one of the two adult U.S. citizens says that Barack Obama was the best president in U.S. history."

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.