Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

4. Probability

Addition Rule

Statistics

4. Probability

Addition Rule

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2:57 minutes

Problem 3.RE.28

Textbook Question

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.

Verified step by step guidance

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.

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.

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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Related Practice

Textbook Question

26. Eye Survey The table shows the results of a survey that asked 3203 people whether they wore contacts or glasses. A person is selected at random from the sample. Find the probability of each event.

a. The person wears only contacts or only glasses.

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Textbook Question

d. The person is male or does not wear glasses.

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Textbook Question

29. Explain, in your own words, why in the Addition Rule for P(A or B or C), P(A and B and C) is added at the end of the formula.

Textbook Question

In Exercises 25 and 26, determine whether the events are mutually exclusive. Explain your reasoning.

25. Event A: Randomly select a red jelly bean from a jar.

Event B: Randomly select a yellow jelly bean from the jar.

Textbook Question

In Exercises 29-32, find the probability.

31. A 12-sided die, numbered 1 to 12, is rolled. Find the probability that the roll results in an odd number or a number less than 4.

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

39. You are given that P(A) = 0.15 and P(B) = 0.40. Do you have enough information to find P(A or B)? Explain.

Textbook Question

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)

a. Randomly selecting someone who is under 10 years old

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