Textbook Question
1. When you calculate the number of permutations of n distinct objects taken r at a time, what are you counting? Give an example.
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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 3.3.13
13. Students A physics class has 40 students. Of these, 12 students are physics majors and 16 students are minoring in math. Of the physics majors, three are minoring in math. Find the probability that a randomly selected student is minoring in math or a physics major.
Verified step by step guidance1
Step 1: Identify the relevant sets and their sizes. The total number of students is 40. The number of physics majors is 12, the number of students minoring in math is 16, and the number of students who are both physics majors and minoring in math is 3.
Step 2: Use the formula for the union of two sets to calculate the probability. The formula is: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( A \) is the event of being a physics major and \( B \) is the event of minoring in math.
Step 3: Calculate \( P(A) \), the probability of being a physics major. This is given by \( \frac{\text{Number of physics majors}}{\text{Total number of students}} \), which is \( \frac{12}{40} \).
Step 4: Calculate \( P(B) \), the probability of minoring in math. This is given by \( \frac{\text{Number of students minoring in math}}{\text{Total number of students}} \), which is \( \frac{16}{40} \).
Step 5: Calculate \( P(A \cap B) \), the probability of being both a physics major and minoring in math. This is given by \( \frac{\text{Number of students who are both}}{\text{Total number of students}} \), which is \( \frac{3}{40} \). Substitute these values into the union formula to find \( P(A \cup B) \).
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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 involves calculating the chance of selecting a student who is either a physics major or minoring in math from a total group of students. Understanding how to compute probabilities is essential for solving the given question.
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Union of Events
The union of two events refers to the occurrence of at least one of the events. In this case, we are interested in the union of the events 'being a physics major' and 'minoring in math.' The formula for the probability of the union of two events is P(A ∪ B) = P(A) + P(B) - P(A ∩ B), which accounts for any overlap between the two groups.
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Set Theory
Set theory provides a framework for understanding collections of objects, which in this case are the students categorized by their academic focus. By using set notation, we can define the groups of physics majors and math minors, and analyze their intersections and unions. This helps in visualizing and calculating the probabilities of the combined events effectively.
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Related Practice
Textbook Question
"Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
12. Not putting money in a parking meter and getting a parking ticket"
Textbook Question
"2. Give an example of
a. two events that are independent.
b. two events that are dependent."
Textbook Question
"Classifying Events Based on Studies In Exercises 15-18, identify the two events described in the study. Do the results indicate that the events are independent or dependent? Explain your reasoning.
15. A study was conducted to debunk the idea that abilities in music and math are related. Instead, the study showed a strong relationship between achievements in music and math.
(Source: University of Kansas)"
Textbook Question
Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
11. Returning a rented movie after the due date and receiving a late fee
Textbook Question
"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
22.Pickup Trucks In a survey, 510 U.S. adults were asked whether they drive a pickup truck and whether they drive a Ford. The results showed that three in twenty adults surveyed drive a Ford. Of the adults surveyed that drive Fords, nine in twenty drive a pickup truck. Find the probability that a randomly selected adult drives a Ford and drives a pickup truck.