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Chapter 5: Problem 85

Consider a chance experiment that consists of selecting a customer at random from all people who purchased a car at a large car dealership during 2016 . a. In the context of this chance experiment, give an example of two events that would be mutually exclusive. b. In the context of this chance experiment, give an example of two events that would not be mutually exclusive.

Short Answer

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Mutually exclusive events example: A customer purchases a sedan car (event A) or a truck (event B). These events are mutually exclusive because a customer cannot purchase both a sedan and a truck at the same time. Non-mutually exclusive events example: A customer purchases a red-colored car (event C) or a car with a sunroof (event D). These events are not mutually exclusive because a customer can purchase a red-colored car with a sunroof, so these events can occur together.

Step by step solution

01

Mutually exclusive events example

Example: Consider two events A and B. Event A represents customers who purchased a sedan car, and event B represents customers who purchased a truck. These two events are mutually exclusive because a customer can either purchase a sedan car or a truck, but not both at the same time. For non-mutually exclusive events, we can consider:
02

Non-mutually exclusive events example

Example: Consider two events C and D. Event C represents customers who purchased a red-colored car, and event D represents customers who purchased a car with a sunroof. These two events are not mutually exclusive because a customer can purchase a red-colored car with a sunroof. So, these events can occur together.

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Key Concepts

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

Mutually Exclusive Events
Mutually exclusive events are events that cannot happen at the same time. This means there is no overlap between these events. Imagine you have two boxes that don't share any items. That's how mutually exclusive events work.
In probability terms, if event A occurs, then event B cannot. The common real-world example of this would be flipping a coin. When you flip a coin, it can either land on heads or tails, but never both heads and tails in one single flip.
In the context of the car dealership example, an event could be a customer purchasing a sedan (Event A) while another event is purchasing a truck (Event B). Since a customer cannot purchase both a sedan and a truck simultaneously, these are mutually exclusive events.
  • Important Fact: The probability of two mutually exclusive events happening at the same time is zero.
Non-Mutually Exclusive Events
Non-mutually exclusive events are events that can occur at the same time. Imagine two overlapping circles in a Venn diagram; the area of overlap represents occurrences where both events happen together.
These events often share some common ground. For instance, a customer purchasing a red car and a customer purchasing a car with a sunroof at the dealership are two events that can overlap because a red car can also have a sunroof.
The real-world analogy could be wearing sunglasses and a hat at the same time; nothing stops these two events from happening together. In probability discussions, understanding non-mutually exclusive events helps in calculating chances where intersection or joint probabilities exist.
  • Remember: The probability of non-mutually exclusive events is the sum of the probabilities of each event minus the intersection of both events.
Chance Experiment
A chance experiment is any situation in which we observe outcomes that occur randomly. Think of it as a setting where you roll dice or flip a coin, where the result is uncertain until it happens.
This concept is essential in probability since it provides the environment to study and calculate the likelihood of certain results.
For example, selecting a customer who purchased a car from the dealership constitutes a chance experiment. Each selection is random, and each customer has an equal probability of being chosen. This randomness is what makes it a 'chance' experiment.
  • Essential Point: Every chance experiment has a sample space, which includes all possible outcomes.
  • Tip: Understanding the setup of a chance experiment is crucial for successfully calculating event probabilities.

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Most popular questions from this chapter

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)?

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.

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 \(P(\underline{C})=\) probability value.

Eighty-six countries won medals at the 2016 Olympics in Rio de Janeiro. Based on the results: 1 country won more than 100 medals 2 countries won between 51 and 100 medals 3 countries won between 31 and 50 medals 4 countries won between 21 and 30 medals 15 countries won between 11 and 20 medals 15 countries won between 6 and 10 medals 46 countries won between 1 and 5 medals Suppose one of the 86 countries winning medals at the 2016 Olympics is selected at random. a. What is the probability that the selected country won more than 50 medals? b. What is the probability that the selected country did not win more than 100 medals? c. What is the probability that the selected country won 10 or fewer medals? d. What is the probability that the selected country won between 11 and 50 medals?

The paper "Predictors of Complementary Therapy Use Among Asthma Patients: Results of a Primary Care Survey" (Health and Social Care in the Community [2008]: \(155-164)\) described a study in which each person in a large sample of asthma patients responded to two questions: Question 1: Do conventional asthma medications usually help your symptoms? Question 2: Do you use complementary therapies (such as herbs, acupuncture, aroma therapy) in the treatment of your asthma? Suppose that this sample is representative of asthma patients. Consider the following events: \(E=\) event that the patient uses complementary therapies \(F=\) event that the patient reports conventional medications usually help The data from the sample were used to estimate the following probabilities: $$ P(E)=0.146 \quad P(F)=0.879 \quad P(E \cap F)=0.122 $$ 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 an asthma patient responds that conventional medications do not help and that the patient uses complementary therapies. ii. The probability that an asthma patient responds that conventional medications do not help and that the patient does not use complementary therapies. iii. The probability that an asthma patient responds that conventional medications usually help or the patient uses complementary therapies. c. Are the events \(E\) and \(F\) independent? Explain.
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