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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.RE.28
Chapter 3, Problem 3.RE.28

28. A sample of 6500 automobiles found that 1560 of the automobiles were black, 3120 of the automobiles were sedans, and 1170 of the automobiles were black sedans. Find the probability that a randomly chosen automobile from this sample is black or a sedan.

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Step 1: Understand the problem. We are tasked with finding the probability that a randomly chosen automobile is either black or a sedan. This involves using the formula for the probability of the union of two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where A represents 'black automobiles' and B represents 'sedans'.
Step 2: Calculate P(A), the probability of an automobile being black. Divide the number of black automobiles (1560) by the total number of automobiles (6500). The formula is P(A) = 1560 / 6500.
Step 3: Calculate P(B), the probability of an automobile being a sedan. Divide the number of sedans (3120) by the total number of automobiles (6500). The formula is P(B) = 3120 / 6500.
Step 4: Calculate P(A ∩ B), the probability of an automobile being both black and a sedan. Divide the number of black sedans (1170) by the total number of automobiles (6500). The formula is P(A ∩ B) = 1170 / 6500.
Step 5: Use the formula for the union of two events to find P(A ∪ B). Substitute the values of P(A), P(B), and P(A ∩ B) into the formula: P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This will give the probability that a randomly chosen automobile is either black or a sedan.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps determine the chance of selecting an automobile that is either black or a sedan from the sample. Understanding how to calculate probabilities is essential for solving the question.
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Introduction to Probability

Union of Events

The union of two events, denoted as A ∪ B, represents the occurrence of at least one of the events. In this case, we are interested in the probability of selecting an automobile that is either black or a sedan. The formula for the union of two events accounts for any overlap, ensuring that we do not double-count automobiles that are both black and sedans.
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Probability of Multiple Independent Events

Counting Principles

Counting principles, such as the addition rule, are used to determine the total number of favorable outcomes when dealing with multiple categories. In this scenario, we need to count the number of black automobiles, sedans, and subtract the overlap (black sedans) to find the total number of automobiles that are either black or sedans. This is crucial for accurately calculating the probability.
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Fundamental Counting Principle
Related Practice
Textbook Question

In Exercises 7-12, classify the statement as an example of classical probability, empirical probability, or subjective probability. Explain your reasoning.

11. The probability of rolling 2 six-sided dice and getting a sum of 9 is 1/9.

Textbook Question

In Exercises 33 and 34, use the pie chart at the left, which shows the percent distribution of the number of students in U.S. public schools in a recent year. (Source: U.S. National Center for Education Statistics)

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34. Find the probability of randomly selecting a school with 300 or more students.

Textbook Question

"In Exercises 17 and 18, use the table, which shows the numbers of first-time and repeat U.S. nursing students taking the National Council Licensure Examination (NCLEX-RN® exam) to pass or fail in a recent year. (Adapted from National Council Licensure Examinations)

18. Find the probability that a student passed, given that the student repeated the exam."

Textbook Question

In Exercises 5 and 6, use the Fundamental Counting Principle.

6. The state of Virginia's license plates have three letters and four digits. Assuming that any letter or digit can be used, how many different license plates are possible?

Textbook Question

"In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.

20. Selecting an ace from a standard deck of 52 playing cards, and then selecting a jack from the deck without replacing the ace"

Textbook Question

In Exercises 13 and 14, use the table, which shows the approximate distribution of the sizes of firms for a recent year. (Adapted from North American Industry Classification System)

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13. Find the probability that a randomly selected firm will have more than four employees.