/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 A small college has 2700 student... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 22

A small college has 2700 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
a. Not mutually exclusive, as a senior can major in computer science. b. Not mutually exclusive, as a female student can major in computer science. c. Mutually exclusive, as a college dormitory resident's residence can't be more than 10 miles away from the campus. d. Depends on the college's policy regarding football team gender restrictions. Additional information is needed for a definite answer.

Step by step solution

01

a. Senior & Majoring in Computer Science

The event that the selected student is a senior, and the event that the selected student is majoring in computer science can both occur at the same time, as it is possible that a senior could be majoring in computer science. Therefore, these events are not mutually exclusive.
02

b. Female & Majoring in Computer Science

The event that the selected student is female, and the event that the selected student is majoring in computer science can both occur at the same time since there is no restriction on gender for majoring in computer science. Therefore, these events are not mutually exclusive.
03

c. College residence > 10 miles & lives in college dormitory

The event that the selected student's college residence is more than 10 miles from the campus, and the event that the selected student lives in a college dormitory cannot both occur at the same time. This is because if a student lives in a college dormitory, their residence can't be more than 10 miles away from the campus. Therefore, these events are mutually exclusive.
04

d. Female & on the college football team

Whether these events are mutually exclusive or not depends on the college's policy. Some colleges have football teams that are open to both male and female students; in such cases, these events are not mutually exclusive. However, if the football team is male-only, these events would be mutually exclusive as a female student can't be on the male-only football team. Additional information about the college's policy is needed to provide a definite answer for this pair of events.

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Theory
At the heart of analyzing scenarios like our textbook exercise is probability theory, which is a branch of mathematics that deals with the likelihood of different outcomes in random events. An understanding of probability theory guides us through the decision-making process under uncertainty. When we talk about events in probability, we're referring to outcomes or occurrences that can happen as a result of an experiment or process.

  • In our exercise, each event is possible: being a senior, majoring in computer science, being female, living more than 10 miles from campus, living in college dormitory, and being on the football team.
  • The concept of mutually exclusive events is critical in probability. Two events are mutually exclusive if they cannot occur at the same time. Think of them as two paths that never cross.

Using probability theory, we can analyze situations to determine if events are mutually exclusive, which has direct applications in fields as varied as managing college demographics or programming artificial intelligence systems.
Random Selection
Random selection is a key principle in statistical sampling and probability, used to ensure that samples represent a population fairly and that all individuals have an equal chance of being chosen. It is a cornerstone of unbiased decision-making in probability.

In our textbook problem, random selection involves choosing a college student without any preference or bias鈥攅ach of the 2700 students has an equal opportunity to be selected.
  • The process is akin to drawing names from a hat, where each name has one entry.
  • This ensures that when we discuss the likelihood of a student being a senior or majoring in computer science, we're considering the entire student body equally.
The accidents of birth, such as gender or residence, should not influence the random selection, which allows probability calculations to be fair and representative of the overall group.
College Demographics
College demographics refer to the statistical characteristics of the student population, such as gender distribution, program enrollment (like computer science majors), year of study (like seniors), and living arrangements (on-campus dormitories versus off-campus housing). These demographics help create diverse and enriching learning environments but also have implications for statistical analysis and probability.

  • For example, if we know that 60% of the student population is female, we can better understand the likelihood of randomly selecting a female student.
  • Additionally, if a small percentage of students are computer science majors, the chances of selecting a senior majoring in this field is lower compared to any senior.

Understanding these demographics is crucial when assessing the probability of events and whether or not they are mutually exclusive, as seen in the textbook exercise. It helps us to explain and predict the dynamics within the student population.

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 rental car company offers two options when a car is rented. A renter can choose to pre-purchase gas or not and can also choose to rent a GPS device or not. Suppose that the events \(A=\) event that gas is pre-purchased \(B=\) event that a GPS is rented are independent with \(P(A)=0.20\) and \(P(B)=0.15\). a. Construct a hypothetical 1000 table with columns corresponding to whether or not gas is pre-purchased and rows corresponding to whether or not a GPS is rented. b. Use the table to find \(P(A \cup B)\). Give a long-run relative frequency interpretation of this probability.

A professor assigns five problems to be completed as homework. At the next class meeting, two of the five problems will be selected at random and collected for grading. You have only completed the first three problems. a. What is the sample space for the chance experiment of selecting two problems at random? (Hint: You can think of the problems as being labeled \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},\) and \(\mathrm{E}\). One possible selection of two problems is \(\mathrm{A}\) and \(\mathrm{B}\). If these two problems are selected and you did problems \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\), you will be able to turn in both problems. There are nine other possible selections to consider.) b. Are the outcomes in the sample space equally likely? c. What is the probability that you will be able to turn in both of the problems selected? d. Does the probability that you will be able to turn in both problems change if you had completed the last three problems instead of the first three problems? Explain. e. What happens to the probability that you will be able to turn in both problems selected if you had completed four of the problems rather than just three?

A deck of 52 cards is mixed well, and 5 cards are dealt. a. It can be shown that (disregarding the order in which the cards are dealt) there are 2,598,960 possible hands, of which only 1287 are hands consisting entirely of spades. What is the probability that a hand will consist entirely of spades? What is the probability that a hand will consist entirely of a single suit? b. It can be shown that exactly 63,206 of the possible hands contain only spades and clubs, with both suits represented. What is the probability that a hand consists entirely of spades and clubs with both suits represented?

Roulette is a game of chance that involves spinning a wheel that is divided into 38 segments of equal size, as shown in the accompanying picture. A metal ball is tossed into the wheel as it is spinning, and the ball eventually lands in one of the 38 segments. Each segment has an associated color. Two segments are green. Half of the other 36 segments are red, and the others are black. When a balanced roulette wheel is spun, the ball is equally likely to land in any one of the 38 segments. a. When a balanced roulette wheel is spun, what is the probability that the ball lands in a red segment? b. In the roulette wheel shown, black and red segments alternate. Suppose instead that all red segments were grouped together and that all black segments were together. Does this increase the probability that the ball will land in a red segment? Explain. c. Suppose that you watch 1000 spins of a roulette wheel and note the color that results from each spin. What would be an indication that the wheel was not balanced?

5.57 There are two traffic lights on Shelly's route from home to work. Let \(E\) denote the event that Shelly must stop at the first light, and define the event \(F\) in a similar manner for the second light. Suppose that \(P(E)=0.4, P(F)=0.3\) and \(P(E \cap F)=0.15 .\) In Exercise \(5.25,\) you constructed a hypothetical 1000 table to calculate the following probabilities. Now use the probability formulas of this section to find these probabilities. a. The probability that Shelly must stop for at least one light (the probability of the event \(E \cup F\) ). b. The probability that Shelly does not have to stop at either light. c. The probability that Shelly must stop at exactly one of the two lights. d. The probability that Shelly must stop only at the first light.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.