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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.

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—each 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.