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Chapter 5: Problem 29

Mutually Exclusive Suppose a student is selected at random in a college. Label each pair of events as mutually exclusive or not mutually exclusive. a. The student studies economics; the student studies statistics. b. The student is pursuing a graduate degree; the student is pursuing a post- graduate degree.

Short Answer

Expert verified
For the given pairs, the first pair of events (studying economics and statistics) is not mutually exclusive, and the second pair of events (pursuing a graduate degree and post-graduate degree) is mutually exclusive.

Step by step solution

01

Analysing first pair of events

For the first pair - The student studies economics; The student studies statistics - these events are not mutually exclusive. A student can be studying both economics and statistics at the same time. These two events/studies don't exclude each other, meaning that both can occur simultaneously.
02

Analysing second pair of events

For the second pair - The student is pursuing a graduate degree; The student is pursuing a post-graduate degree - these events are mutually exclusive. A student can't be pursuing a graduate degree and a postgraduate degree at the same time. In academic progression, a postgraduate degree typically follows graduate studies. Hence, these two events can't occur at the same time.

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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 statistics lies probability theory, a mathematical framework designed for studying uncertainty and randomness. It revolves around the concept of probability, which quantifies how likely an event is to occur. Within this framework, the notion of mutually exclusive events is fundamental.

Mutually exclusive events are events that cannot occur at the same time. Think of a coin flip 鈥� it can't be both heads and tails on the same flip. When considering whether events are mutually exclusive in probability, it鈥檚 essential to understand that if two events are mutually exclusive, the probability of both occurring simultaneously is zero.
Statistics Education
In the realm of statistics education, comprehending the concept of mutually exclusive events is vital. Educators must ensure students grasp the defintion and implications of such events in practical scenarios. For example, when a student selects courses, they need to realize that certain classes may be scheduled at the same time, making enrollment in both mutually exclusive.

Educational strategies, like real-life examples and step-by-step problem solving as seen in the given exercise, can significantly aid in understanding. Using visual aids like Venn diagrams or probability trees can also elucidate these concepts, making the abstract notions more tangible for learners.
Statistical Concepts
Delving deeper into statistical concepts, it's crucial to distinguish between mutually exclusive and independent events, as they're often confused. Independence refers to the scenario where the occurrence of one event does not impact the likelihood of another. For instance, the weather being sunny doesn't affect the probability of you drawing a red card from a deck.

In contrast, mutually exclusive events are tied by the impossibility of their co-occurrence, which directly influences the calculation of probabilities. This distinction helps in solving various statistical problems, including those that determine the likelihood of events in academics, like the progression from graduate to post-graduate studies.

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Most popular questions from this chapter

Red Light/Green Light A busy street has three traffic lights in a row. These lights are not synchronized, so they run independently of each other. At any given moment, the probability that a light is green is \(60 \%\). Assuming that there is no traffic, follow the steps below to design a simulation to estimate the probability that you will get three green lights. a. Identify the action with a random outcome, and give the probability that it is a success. b. Explain how you will simulate this action using the random number table in Appendix A. Which digits will represent green and which nongreen? If you want to get the same results we did, use all of the possible one-digit numbers \((0,1,2,3,4,5,6,7,8\), and 9\()\), and let the first few represent the green lights. How many and what numbers would represent green lights, and what numbers would represent non-green lights? c. Describe the event of interest. d. Explain how you will simulate a single trial. e. Carry out 20 repetitions of your trial, beginning with the first digit on line 11 of the random number table. For each trial, list the random digits, the outcomes they represent, and whether or not the event of interest happened. f. What is the empirical probability that you get three green lights?

Empirical vs. Theoretical A friend flips a coin 10 times and says that the probability of getting a head is \(60 \%\) because he got six heads. Is the friend referring to an empirical probability or a theoretical probability? Explain.

Mutually Exclusive Suppose a person is selected at random. Label each pair of events as mutually exclusive or not mutually exclusive. a. The person has brown eyes; the person has blue eyes. b. The person is 50 years old; the person is a U.S. senator.

Empirical vs. Theoretical A magician claims that he has a fair coin- "fair" because both sides, heads and tails, are equally likely to land face up when the coin is flipped. He tells you that if you flip the coin three times, the probability of getting three tails is \(1 / 8\). Is this an empirical probability or a theoretical probability? Explain.

GPA The probability of a randomly selected person having a GPA of \(8.5\) or above in all subjects is \(0.25\). a. If two students are chosen randomly and independently, what is the probability that they both have a GPA of \(8.5\) or above? b. If two students are selected from the same high school statistics class, do you think the probability of their both having a GPA of \(8.5\) or above is different from your answer to part a? Explain.
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