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 3.2.24a

Chapter 3, Problem 3.2.24a

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)
a. Find the probability that all four have 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 all four randomly selected children have lost a friend or relative to murder. This involves using the Multiplication Rule for independent events, as the probability of one child being affected does not influence the others.

Step 2: Identify the given information. From the problem, we know that 8% of children (or 0.08 as a decimal) have lost a friend or relative to murder. This is the probability of a single child being affected, denoted as P(A) = 0.08.

Step 3: Recognize that the events are independent. Since the selection of one child does not affect the selection of another, the probability of all four children being affected can be calculated by multiplying the probabilities of each child being affected.

Step 4: Apply the Multiplication Rule. The probability of all four children being affected is given by the formula: P(All 4) = P(A) × P(A) × P(A) × P(A), which can also be written as P(All 4) = P(A)^4.

Step 5: Substitute the given probability into the formula. Replace P(A) with 0.08 in the formula P(All 4) = P(A)^4 to calculate the probability. This will give you the final result.

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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 occurring together is the product of their individual probabilities. In this context, if the events are the selection of children who have lost a friend or relative to murder, the rule allows us to calculate the probability of all selected children experiencing this event by multiplying the individual probabilities for each child.

Independent Events

Independent events are those whose outcomes do not affect each other. In this scenario, the selection of one child does not influence the selection of another, assuming the sample size is large enough. This independence is crucial for applying the Multiplication Rule correctly, as it ensures that the probabilities can be multiplied without adjustment.

Probability Calculation

Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. To find the probability that all four selected children have lost a friend or relative to murder, we first determine the probability for one child (8% or 0.08) and then raise this probability to the power of four, reflecting the four independent selections.

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Related Practice

Textbook Question

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

28. Blood Types The probability that a Latinx American person in the United States has type A+ blood is 29%. Four Latinx American people in the United States are selected at random. (Source: American National Red Cross)

a. Find the probability that all four have type A+ blood."

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Textbook Question

Shuffle Play You use a shuffle playback feature to randomly play songs in a playlist. The playlist of 56 songs includes 15 instrumental songs.

a. What is the probability that the first three songs to play are instrumental songs? (Assume a song cannot be repeated.)

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Textbook Question

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

26. Worst President In a sample of 1500 adult U.S. citizens, 690 said that Donald Trump was the worst president in U.S. history. Three adult U.S. citizens are selected at random.

(Adapted from YouGov)

b. Find the probability that none of the three adult U.S. citizens say that Donald Trump was the worst president in U.S. history."

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Textbook Question

Finding Conditional Probabilities In Exercises 7 and 8, use the table to find each conditional probability.

7. Business Degrees The table shows the numbers of male and female students in the United States who received bachelor's degrees in business and nonbusiness fields in a recent year. (Source: National Center for Educational Statistics)

b. Find the probability that a randomly selected bachelor's degree-earning student received a business degree, given that the student is female.

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Textbook Question

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

29. In Vitro Fertilization In a recent year, about 1.9% of all infants born in the U.S. were conceived through assisted reproductive technology (ART). Of the ART deliveries, about 26.4% resulted in multiple births. (Source: Morbidity and Mortality Weekly Report)

a. Find the probability that a randomly selected infant was conceived through ART and was part of a multiple birth.

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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)

a. Find the probability that all six have type O+ blood."

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