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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2:55 minutes

Problem 3.RE.31

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.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability of rolling an odd number OR a number less than 4 on a 12-sided die. This involves calculating the probability of two events and combining them using the formula for the union of two events.

Step 2: Define the events. Let Event A represent rolling an odd number, and Event B represent rolling a number less than 4. The die is numbered from 1 to 12, so the sample space consists of these 12 outcomes.

Step 3: Identify the outcomes for each event. For Event A (odd numbers), the outcomes are {1, 3, 5, 7, 9, 11}. For Event B (numbers less than 4), the outcomes are {1, 2, 3}.

Step 4: Determine the union of the two events. The union includes all outcomes that are either odd or less than 4. Combine the outcomes from Event A and Event B, ensuring no duplicates. The union is {1, 2, 3, 5, 7, 9, 11}.

Step 5: Calculate the probability. The probability of an event is the number of favorable outcomes divided by the total number of outcomes in the sample space. Count the outcomes in the union (7 outcomes) and divide by the total number of outcomes (12 outcomes). Use the formula: \( P(A \cup B) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \).

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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. An event with a probability of 0 means it cannot happen, while a probability of 1 means it is certain to happen. In this context, we calculate the probability of rolling an odd number or a number less than 4 on a 12-sided die.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For a 12-sided die, the sample space consists of the numbers 1 through 12. Understanding the sample space is crucial for calculating probabilities, as it provides the total number of outcomes against which specific events can be measured.

Union of Events

The union of events refers to the occurrence of at least one of the events in a set. In this case, we are interested in the union of rolling an odd number and rolling a number less than 4. To find the probability of the union, we must consider the individual probabilities of each event and subtract any overlap to avoid double counting.

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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.

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

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.

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

Textbook Question

Politics The responses of 1500 U.S. adults to a survey that asked them to state their own political viewpoints are shown in the Pareto chart. Find the probability of each event.(Adapted from YouGov)

b. Randomly selecting a person from the sample who is conservative or very conservative

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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.

Key Concepts

Probability

Sample Space

Union of Events

Watch next

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.