Textbook Question
In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
22. Getting high grades and being awarded an academic scholarship
Ch. 3 - Probability
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 3, Problem 3.RE.31
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 guidance1
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.
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Introduction to Probability
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.
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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
In Exercises 7-12, classify the statement as an example of classical probability, empirical probability, or subjective probability. Explain your reasoning.
8. The probability of randomly selecting five cards of the same suit from a standard deck of 52 playing cards is about 0.002.
Textbook Question
You work in the security department of a bank’s website. To access their accounts, customers of the bank must create an 8-digit password. It is your job to determine the password requirements for these accounts. Security guidelines state that for the website to be secure, the probability that an 8-digit password is guessed on one try must be less than 1/60^8, assuming all passwords are equally likely.
Your job is to use the probability techniques you have learned in this chapter to decide what requirements a customer must meet when choosing a password, including what sets of characters are allowed, so that the website is secure according to the security guidelines.
3. For additional security, each customer creates a 5-digit PIN (personal identification number). The table on the right shows the 10 most commonly chosen 5-digit PINs. From the table, you can see that more than a third of all 5-digit PINs could be guessed by trying these 10 numbers. To discourage customers from using predictable PINs, you consider prohibiting PINs that use the same digit more than once.
b. Would you decide to prohibit PINs that use the same digit more than once? Explain.
Textbook Question
You work in the security department of a bank’s website. To access their accounts, customers of the bank must create an 8-digit password. It is your job to determine the password requirements for these accounts. Security guidelines state that for the website to be secure, the probability that an 8-digit password is guessed on one try must be less than 1/60^8, assuming all passwords are equally likely.
Your job is to use the probability techniques you have learned in this chapter to decide what requirements a customer must meet when choosing a password, including what sets of characters are allowed, so that the website is secure according to the security guidelines.
2. Answering the Question
a. What password requirements would you set? What characters would be allowed?
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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
In Exercises 35–38, the bar graph shows the results of a survey in which 8806 undergraduate students were asked how many hours they spend on studying and other academic activities outside of class in a typical week. (Source: American College Health Association)
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37. Find the probability of randomly selecting an undergraduate who does not study from 6 to 10 hours per week.