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 3.2.25c

Chapter 3, Problem 3.2.25c

"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."

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that at least one of the two randomly selected adult U.S. citizens says that Barack Obama was the best president in U.S. history. This involves using the complement rule and the multiplication rule.

Step 2: Calculate the probability that a single adult U.S. citizen says Barack Obama was the best president. This is given by dividing the number of favorable outcomes (270) by the total sample size (1500). The formula is:

\frac{270}{1500}

Step 3: Calculate the probability that a single adult U.S. citizen does NOT say Barack Obama was the best president. This is the complement of the probability calculated in Step 2. The formula is:

1 - \frac{270}{1500}

Step 4: Use the multiplication rule to find the probability that neither of the two selected citizens says Barack Obama was the best president. Multiply the probability from Step 3 by itself (since the selections are independent). The formula is:

(1 - \frac{270}{1500}) \times (1 - \frac{270}{1500})

Step 5: Use the complement rule to find the probability that at least one of the two citizens says Barack Obama was the best president. Subtract the result from Step 4 from 1. The formula is:

1 - (1 - \frac{270}{1500}) \times (1 - \frac{270}{1500})

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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 occurring together is the product of their individual probabilities. In this context, it helps calculate the likelihood of multiple selections from a population, allowing us to determine the combined probability of outcomes when selecting two individuals.

Probability of Complement

The probability of the complement of an event is the likelihood that the event does not occur. In this question, to find the probability that at least one of the two selected citizens believes Barack Obama is the best president, we can first calculate the probability that neither does and then subtract that from 1.

Sample Proportion

Sample proportion refers to the ratio of individuals in a sample who exhibit a certain characteristic to the total number of individuals in that sample. In this case, the sample proportion of citizens who think Barack Obama is the best president is 270 out of 1500, which is essential for calculating the probabilities involved in the question.

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Sampling Distribution of Sample Proportion

Related Practice

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

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

23. Celebrities as Role Models In a sample of 1103 probable voters, three out of four say they would like entertainers to address social and political issues. Two probable voters are selected at random. (Source: The Hollywood Reporter)

c. Find the probability that at least one of the two probable voters would like entertainers to address social and political issues."

Textbook Question

19. U.S. Age Distribution The projected percent distribution of the U.S. population for 2025 is shown in the pie chart. Find the probability of each event. (Source: U.S. Census

Bureau)

c. Randomly selecting someone who is not 60 years or over

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

23. Engineering Degrees The table shows the numbers of male and female students in the U.S. who received B.S. degrees in engineering in a recent year. A student earning a B.S. degree in engineering during that year is selected at random. Find the probability of each event.

(Source: National Center for Educational Statistics)

c. The student is not female or did not receive a mechanical engineering degree.