Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 5: Problem 22

A small college has 2,700 students enrolled. Consider the chance experiment of selecting a student at random. For each of the following pairs of events, indicate whether or not you think they are mutually exclusive and explain your reasoning. a. the event that the selected student is a senior and the event that the selected student is majoring in computer science. b. the event that the selected student is female and the event that the selected student is majoring in computer science. c. the event that the selected student's college residence is more than 10 miles from campus and the event that the selected student lives in a college dormitory. d. the event that the selected student is female and the event that the selected student is on the college football team.

Short Answer

Expert verified
Event pairs a and b are not mutually exclusive, while event pair c is mutually exclusive. Event pair d is not necessarily mutually exclusive without additional information about the college's sports teams.

Step by step solution

01

Analyze event pair a

Consider the events 'the selected student is a senior' and 'the selected student is majoring in computer science'. A student can be a senior as well as major in computer science simultaneously. Therefore, these events are not mutually exclusive.
02

Analyze event pair b

Consider the events 'the selected student is female' and 'the selected student is majoring in computer science'. A student can be female and major in computer science at the same time. Hence, these events are not mutually exclusive.
03

Analyze event pair c

The events 'the selected student's college residence is more than 10 miles from campus' and 'the selected student lives in a college dormitory' are mutually exclusive. The reason is that a student can't live more than 10 miles away from the college and simultaneously live in a college dormitory.
04

Analyze event pair d

Evaluate the events 'the selected student is female' and 'the selected student is on the college football team'. Depending on the college, females may or may not be part of the football team. Without additional information, we should not assume these events are mutually exclusive.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
Understanding probability theory is essential when dealing with the concept of mutually exclusive events. Probability theory is the branch of mathematics that deals with the likelihood of certain outcomes occurring within a given set of circumstances. In the context of our exercise, we focus on the probability of selecting a student from a small college, based on certain characteristics or events.

When two events cannot happen at the same time, they are considered to be mutually exclusive. For instance, a student cannot simultaneously be in their first year of college (a freshman) and in their fourth year (a senior). Analyzing whether events are mutually exclusive helps us make logical conclusions about their probabilities. The addition rule of probability helps to calculate the probability of either of two mutually exclusive events occurring, which is simply the sum of their individual probabilities.
College Student Demographics
College student demographics refer to the statistical data reflecting the characteristics of a student population. This includes information such as age, gender, grade level, major, distance from home to the college, and residence status. Understanding demographic trends is crucial for colleges to support their diverse student bodies.

Diversity in college campuses can influence the possibility of events being mutually exclusive. For instance, in exercise step b, we considered if being female and majoring in computer science were mutually exclusive. The demographics showing an increasing number of females in STEM fields would support that these two events are, in fact, not mutually exclusive. Demographics can provide essential context that affects our analysis of certain events.
Statistical Reasoning
Statistical reasoning involves making sense of data by applying statistical principles and concepts. When evaluating our mutually exclusive events, we relied on statistical reasoning to inform our decisions. In each step of the provided solution, we used logical analysis grounded in an understanding of what it means for events to occur concurrently.

For example, it is statistically reasonable to conclude that a student cannot both live in a dormitory and more than 10 miles from campus. This analysis is based on a proper understanding of the conditions and constraints of college living arrangements. Throughout these exercises, statistical reasoning is applied to infer whether the given pairs of events are mutually exclusive, leading to more robust and logically sound conclusions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A student placement center has requests from five students for employment interviews. Three of these students are math majors, and the other two students are statistics majors. Unfortunately, the interviewer has time to talk to only two of the students. These two will be randomly selected from among the five. a. What is the sample space for the chance experiment of selecting two students at random? (Hint: You can think of the students as being labeled \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and \(\mathrm{E}\). One possible selection of two students is \(\mathrm{A}\) and \(\mathrm{B}\). There are nine other possible selections to consider.) b. Are the outcomes in the sample space equally likely? c. What is the probability that both selected students are statistics majors? d. What is the probability that both students are math majors? e. What is the probability that at least one of the students selected is a statistics major? f. What is the probability that the selected students have different majors?

Suppose you want to estimate the probability that a patient will develop an infection while hospitalized at a particular hospital. In the past year, this hospital had 6,450 patients, and 712 of them developed an infection. What is the estimated probability that a patient at this hospital will develop an infection?

A large cable TV company reports the following: \- \(80 \%\) of its customers subscribe to its cable TV service \- \(42 \%\) of its customers subscribe to its Internet service \- \(32 \%\) of its customers subscribe to its telephone service \(25 \%\) of its customers subscribe to both its cable TV and Internet service \(21 \%\) of its customers subscribe to both its cable TV and phone service \- \(23 \%\) of its customers subscribe to both its Internet and phone service \- \(15 \%\) of its customers subscribe to all three services Consider the chance experiment that consists of selecting one of the cable company customers at random. Find and interpret the following probabilities: a. \(P(\) cable TV only \()\) b. \(P(\) Internet \(\mid\) cable \(\mathrm{TV})\) c. \(P\) (exactly two services) d. \(P\) (Internet and cable TV only)

Suppose that \(20 \%\) of all teenage drivers in a certain county received a citation for a moving violation within the past year. Assume in addition that \(80 \%\) of those receiving such a citation attended traffic school so that the citation would not appear on their permanent driving record. Consider the chance experiment that consists of randomly selecting a teenage driver from this county. a. One of the percentages given in the problem specifies an unconditional probability, and the other percentage specifies a conditional probability. Which one is the conditional probability, and how can you tell? b. Suppose that two events \(E\) and \(F\) are defined as follows: \(E=\) selected driver attended traffic school \(F=\) selected driver received such a citation Use probability notation to translate the given information into two probability statements of the form \(P(\ldots)=\) probability value.

Four students must work together on a group project. They decide that each will take responsibility for a particular part of the project, as follows: Because of the way the tasks have been divided, one student must finish before the next student can begin work. To ensure that the project is completed on time, a time line is established, with a deadline for each team member. If any one of the team members is late, the timely completion of the project is jeopardized. Assume the following probabilities: 1\. The probability that Maria completes her part on time is 0.8 2\. If Maria completes her part on time, the probability that Alex completes on time is \(0.9,\) but if Maria is late, the probability that Alex completes on time is only 0.6 . 3\. If Alex completes his part on time, the probability that Juan completes on time is \(0.8,\) but if \(\mathrm{Alex}\) is late, the probability that Juan completes on time is only 0.5 . 4\. If Juan completes his part on time, the probability that Jacob completes on time is \(0.9,\) but if Juan is late, the probability that Jacob completes on time is only 0.7 . Use simulation (with at least 20 trials) to estimate the probability that the project is completed on time. Think carefully about this one. For example, you might use a random digit to represent each part of the project (four in all). For the first digit (Maria's part), \(1-8\) could represent on time, and 9 and 0 could represent late. Depending on what happened with Maria (late or on time), you would then look at the digit representing Alex's part. If Maria was on time, \(1-9\) would represent on time for Alex, but if Maria was late, only \(1-6\) would represent on time. The parts for Juan and Jacob could be handled similarly.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free