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 3.2.26b

Chapter 3, Problem 3.2.26b

"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."

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that none of the three randomly selected adult U.S. citizens say that Donald Trump was the worst president in U.S. history. This involves using the Multiplication Rule for independent events.

Step 2: Calculate the probability that a single adult U.S. citizen does NOT say that Donald Trump was the worst president. From the sample, 690 out of 1500 said he was the worst president. Therefore, the probability that a single person does NOT say this is given by \( P(\text{not worst}) = 1 - \frac{690}{1500} \).

Step 3: Recognize that the three selections are independent events. The probability that none of the three selected individuals say that Donald Trump was the worst president is the product of the probabilities for each individual.

Step 4: Use the Multiplication Rule. Multiply the probability of \( P(\text{not worst}) \) for each of the three individuals: \( P(\text{none say worst}) = P(\text{not worst}) \times P(\text{not worst}) \times P(\text{not worst}) \).

Step 5: Simplify the expression. Substitute the value of \( P(\text{not worst}) \) from Step 2 into the formula from Step 4 and simplify to find 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 of Probability

The Multiplication Rule states that the probability of two or more independent events occurring together is the product of their individual probabilities. In this context, it helps calculate the likelihood of multiple citizens independently expressing the same opinion about Donald Trump being the worst president.

Independent Events

Independent events are those whose outcomes do not affect each other. In this scenario, the selection of one adult citizen does not influence the opinions of the others, allowing us to treat their responses as independent when calculating probabilities.

Complementary Probability

Complementary probability refers to the likelihood of an event not occurring. In this case, to find the probability that none of the three selected citizens say Trump is the worst president, we first determine the probability that one citizen does not hold that opinion and then apply the Multiplication Rule to find the overall probability for three citizens.

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Complementary Events

Related Practice

Textbook Question

18. Rolling a Die You roll a die. Find the probability of each event.

b. Rolling a 2 or an odd number

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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

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.

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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.

b. Find the probability that none of the four have type A+ blood.

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