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Chapter 7: Problem 5

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\) \(\\{b, c, e\\}\) be events of this experiment. Are the events \(E\) and \(F\) mutually exclusive?

Short Answer

Expert verified
The events \(E\) and \(F\) are not mutually exclusive because their intersection, \(E \cap F = \{a\}\), is not an empty set.

Step by step solution

01

Examine the events

The events to be examined are \(E = \{a, b\}\) and \(F = \{a, d, f\}\). These sets represent the outcomes for two different events.
02

Find the intersection of E and F

The intersection of two sets is the set of elements that are common to both sets. To find the intersection of \(E\) and \(F\), look for any elements that appear in both \(E\) and \(F\). In this case, the intersection \(E \cap F = \{a\}\).
03

Determine if E and F are mutually exclusive

Finally, to determine if events \(E\) and \(F\) are mutually exclusive, check if their intersection is an empty set. From the result in Step 2, the intersection of \(E\) and \(F\) is not an empty set (\(\{a\}\) is not empty), therefore events \(E\) and \(F\) are not mutually exclusive. In conclusion, the events \(E\) and \(F\) are not mutually exclusive because they have a common outcome \(a\).

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Key Concepts

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

Probability Theory
Understanding probability theory is crucial when delving into the world of statistical certainty and randomness. It's a mathematical framework for quantifying the likelihood of events occurring within a certain context. In probability theory, every event has a probability value between 0 and 1, with 0 indicating impossibility and 1 signifying certainty.

Various types of events can occur, such as independent events, which have no effect on the likelihood of other events, and mutually exclusive events, which cannot occur simultaneously. For example, if you flip a coin, getting heads and tails at the same time is impossible; these are mutually exclusive outcomes. The concept of mutually exclusive events is crucial in calculating the probability of either event occurring, as their probabilities would be additive if they are mutually exclusive.
Sample Space
The concept of a sample space is a foundational element in probability theory. It is the set of all possible outcomes of a particular experiment. For instance, if you're rolling a six-sided die, the sample space would be \( S=\{1, 2, 3, 4, 5, 6\} \). The size of the sample space is a guide to determining the likelihood of various events.

In a well-defined experiment, it's important to identify the complete sample space before calculating probabilities, as each event is a subset of the sample space. The power of the sample space concept comes into play when assessing complex scenarios, as you can systematically breakdown the list of all potential outcomes to address the problem at hand.
Intersection of Sets
When dealing with sets in probability, the intersection plays a significant role. The intersection of two sets is a new set containing all elements that are present in both sets. It's symbolized by \( \cap \) and can be stated as \( A \cap B \), which reads as 'A intersect B'.

In probability, finding the intersection of two events' sets helps us determine if events are mutually exclusive or not. If the intersection of two events is an empty set, then the events are mutually exclusive, meaning the occurrence of one event rules out the occurrence of the other. In the given exercise, the intersection \( E \cap F = \{a\} \) reveals that the events \( E \) and \( F \) share an outcome, thus they are not mutually exclusive.

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Most popular questions from this chapter

Data compiled by the Department of Justice on the number of people arrested in a certain year for serious crimes (murder, forcible rape, robbery, and so on) revealed that \(89 \%\) were male and \(11 \%\) were female. Of the males, \(30 \%\) were under 18 , whereas \(27 \%\) of the females arrested were under 18 . a. What is the probability that a person arrested for a serious crime in that year was under 18 ? b. If a person arrested for a serious crime in that year is known to be under 18 , what is the probability that the person is female?

In an online survey of 1962 executives from 64 countries conducted by Korn/Ferry International between August and October 2006, the executives were asked if they would try to influence their children's career choices. Their replies: A (to a very great extent), B (to a great extent), \(\mathrm{C}\) (to some extent), D (to a small extent), and \(\mathrm{E}\) (not at all) are recorded below: $$\begin{array}{lccccc} \hline \text { Answer } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 135 & 404 & 1057 & 211 & 155 \\ \hline \end{array}$$ What is the probability that a randomly selected respondent's answer was \(\mathrm{D}\) (to a small extent) or \(\mathrm{E}\) (not at all)?

According to a study of 100 drivers in metropolitan Washington D.C. whose cars were equipped with cameras with sensors, the distractions and the number of incidents (crashes, near crashes, and situations that require an evasive maneuver after the driver was distracted) caused by these distractions are as follows: $$ \begin{array}{lccccccccc} \hline \text { Distraction } & A & B & C & D & E & F & G & H & I \\ \hline \text { Driving Incidents } & 668 & 378 & 194 & 163 & 133 & 134 & 111 & 111 & 89 \\ \hline \end{array} $$ where \(A=\) Wireless device (cell phone, PDA) $$ \begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside } \mathrm{car} \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned} $$ If an incident caused by a distraction is picked at random, what is the probability that it was caused by a. The use of a wireless device? b. Something other than personal hygiene or eating?

Electricity in the United States is generated from many sources. The following table gives the sources as well as their share in the production of electricity: $$ \begin{array}{lcccccc} \hline \text { Source } & \text { Coal } & \text { Nuclear } & \text { Natural gas } & \text { Hydropower } & \text { Oil } & \text { Other } \\ \hline \text { Share, } \% & 50.0 & 19.3 & 18.7 & 6.7 & 3.0 & 2.3 \\ \hline \end{array} $$ If a source for generating electricity is picked at random, what is the probability that it comes from a. Coal or natural gas? b. Nonnuclear sources?

The chief loan officer of La Crosse Home Mortgage Company summarized the housing loans extended by the company in 2007 according to type and term of the loan. Her list shows that \(70 \%\) of the loans were fixed-rate mortgages \((F), 25 \%\) were adjustable-rate mortgages \((A)\), and \(5 \%\) belong to some other category \((O)\) (mostly second trust-deed loans and loans extended under the graduated payment plan). Of the fixed-rate mortgages, \(80 \%\) were 30 -yr loans and \(20 \%\) were 15 -yr loans; of the adjustable-rate mortgages, \(40 \%\) were 30 -yr loans and \(60 \%\) were 15 -yr loans; finally, of the other loans extended, \(30 \%\) were 20 -yr loans, \(60 \%\) were 10 -yr loans, and \(10 \%\) were for a term of 5 yr or less. a. Draw a tree diagram representing these data. b. What is the probability that a home loan extended by La Crosse has an adjustable rate and is for a term of 15 yr? c. What is the probability that a home loan cxtended by La Crosse is for a term of 15 yr?
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