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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.

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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.

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Most popular questions from this chapter

An appliance manufacturer offers extended warranties on its washers and dryers. Based on past sales, the manufacturer reports that of customers buying both a washer and a dryer, \(52 \%\) purchase the extended warranty for the washer, \(47 \%\) purchase the extended warranty for the dryer, and \(59 \%\) purchase at least one of the two extended warranties. a. Use the given probability information to set up a hypothetical 1000 table. b. Use the table from Part (a) to find the following probabilities: i. the probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer. ii. the probability that a randomly selected customer purchases an extended warranty for neither the washer nor the dryer.

A study of the impact of seeking a second opinion about a medical condition is described in the paper "Evaluation of Outcomes from a National Patient- Initiated Second-Opinion Program". Based on a review of 6791 patient-initiated second opinions, the paper states the following: "Second opinions often resulted in changes in diagnosis (14.8\%), treatment \((37.4 \%),\) or changes in both \((10.6 \%)\)." Consider the following two events: \(D=\) event that second opinion results in a change in diagnosis \(T=\) event that second opinion results in a change in treatment a. What are the values of \(P(D), P(T),\) and \(P(D \cap T) ?\) b. Use the given probability information to set up a hypothetical 1000 table with columns corresponding to \(D\) and \(D^{C}\) and rows corresponding to \(T\) and \(T^{C}\). c. What is the probability that a second opinion results in neither a change in diagnosis nor a change in treatment? d. What is the probability that a second opinion results is a change in diagnosis or a change in treatment?

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?

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 \(P(E \cap F)=0.15\). a. Use the given probability information to set up a hypothetical 1000 table with columns corresponding to \(E\) and not \(E\) and rows corresponding to \(F\) and not \(F\). b. Use the table from Part (a) to find the following probabilities: i. the probability that Shelly must stop for at least one light (the probability of \(E \cup F)\). ii. the probability that Shelly does not have to stop at either light. iii. the probability that Shelly must stop at exactly one of the two lights. iv. the probability that Shelly must stop only at the first light.

5.65 The authors of the paper "Do Physicians Know When Their Diagnoses Are Correct?" (Journal of General Internal Medicine [2005]: 334-339) presented detailed case studies to medical students and to faculty at medical schools. Each participant was asked to provide a diagnosis in the case and also to indicate whether his or her confidence in the correctness of the diagnosis was high or low. Define the events \(C\), \(I,\) and \(H\) as follows: \(C=\) event that diagnosis is correct \(I=\) event that diagnosis is incorrect \(H=\) event that confidence in the correctness of the diagnosis is high a. Data appearing in the paper were used to estimate the following probabilities for medical students: $$ \begin{array}{r} P(C)=0.261 \\ P(I)=0.739 \\ P(H \mid C)=0.375 \\ P(H \mid I)=0.073 \end{array} $$ Use Bayes' Rule to calculate the probability of a correct diagnosis given that the student's confidence level in the correctness of the diagnosis is high. b. Data from the paper were also used to estimate the following probabilities for medical school faculty: $$ \begin{array}{r} P(C)=0.495 \\ P(I)=0.505 \\ P(H \mid C)=0.537 \\ P(H \mid I)=0.252 \end{array} $$ Calculate \(P(C \mid H)\) for medical school faculty. How does the value of this probability compare to the value of \(P(C \mid H)\) for students calculated in Part (a)?
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