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.39
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.
Verified step by step guidance1
Step 1: Recall the formula for the probability of the union of two events, P(A or B), which is given by P(A ∪ B) = P(A) + P(B) - P(A ∩ B). This formula accounts for the overlap between events A and B to avoid double-counting.
Step 2: Substitute the given values into the formula. You are provided P(A) = 0.15 and P(B) = 0.40. However, the formula also requires the value of P(A ∩ B), the probability that both events A and B occur simultaneously.
Step 3: Assess whether P(A ∩ B) is provided or can be inferred. In this problem, P(A ∩ B) is not given, and there is no information about whether events A and B are independent or dependent. Without P(A ∩ B), you cannot calculate P(A or B) precisely.
Step 4: If events A and B were independent, you could calculate P(A ∩ B) using the formula P(A ∩ B) = P(A) × P(B). However, independence is not stated in the problem, so you cannot assume it.
Step 5: Conclude that you do not have enough information to find P(A or B) because the value of P(A ∩ B) is missing, and the relationship between events A and B (independent or dependent) is not specified.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability of Events
Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, P(A) and P(B) represent the probabilities of events A and B occurring, respectively. Understanding how to interpret these probabilities is essential for analyzing the relationship between the events.
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Probability of Multiple Independent Events
Union of Events
The union of two events, denoted as P(A or B), represents the probability that at least one of the events occurs. To calculate this, we typically use the formula P(A or B) = P(A) + P(B) - P(A and B). However, without knowing P(A and B), we cannot determine P(A or B) accurately.
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Independence of Events
Events A and B are independent if the occurrence of one does not affect the probability of the other. If A and B are independent, we can calculate P(A and B) as P(A) * P(B). However, the question does not provide information about their independence, which is crucial for determining P(A or B) without additional data.
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Related Practice
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 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
In Exercises 19-22, determine whether the events are independent or dependent. Explain your reasoning.
19. Tossing a coin four times and getting four heads, and then tossing it a fifth time and getting a head
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 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.