Chapter 7

Let \(S\) be a sample space for an experiment. Show that if \(E\) is any event of an experiment, then \(E\) and \(E^{c}\) are mutually exclusive.

How many different signals can be made by hoisting two yellow flags, four green flags, and three red flags on a ship's mast at the same time?

Let \(U\) denote the set of all employees in a hospital. Let $$ \begin{array}{l} \boldsymbol{N}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a nurse }\\} \\ \boldsymbol{D}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a doctor\\} } \\ \boldsymbol{A}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is an administrator\\} } \\ \boldsymbol{M}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a male\\} } \\ \boldsymbol{F}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a female }\\} \end{array} $$ Describe each set in words. a. \(N \cup D\) b. \(N \cap M\)

A poll was conducted among 250 residents of a certain city regarding tougher gun-control laws. The results of the poll are shown in the table: $$ \begin{array}{lccccc} \hline & \begin{array}{c} \text { Own } \\ \text { Only a } \\ \text { Handgun } \end{array} & \begin{array}{c} \text { Own } \\ \text { Only a } \\ \text { Rifle } \end{array} & \begin{array}{c} \text { Own a } \\ \text { Handgun } \\ \text { and a Rifle } \end{array} & \begin{array}{c} \text { Own } \\ \text { Neither } \end{array} & \text { Total } \\ \hline \text { Favor } & & & & & \\ \text { Tougher Laws } & 0 & 12 & 0 & 138 & 150 \\ \hline \begin{array}{l} \text { Oppose } \\ \text { Tougher Laws } \end{array} & 58 & 5 & 25 & 0 & 88 \\ \hline \text { No } & & & & & \\ \text { Opinion } & 0 & 0 & 0 & 12 & 12 \\ \hline \text { Total } & 58 & 17 & 25 & 150 & 250 \\ \hline \end{array} $$ If one of the participants in this poll is selected at random, what is the probability that he or she a. Favors tougher gun-control laws? b. Owns a handgun? c. Owns a handgun but not a rifle? d. Favors tougher gun-control laws and does not own a handgun?

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}\right\\}\) be the sample space associated with an experiment having the following probability distribution: $$ \begin{array}{lcccccc} \hline \text { Outcome } & s_{1} & s_{2} & s_{3} & s_{4} & s_{5} & s_{6} \\ \hline \text { Probability } & \frac{1}{12} & \frac{1}{4} & \frac{1}{12} & \frac{1}{6} & \frac{1}{3} & \frac{1}{12} \\ \hline \end{array} $$ Find the probability of the event: a. \(A=\left\\{s_{1}, s_{3}\right\\}\) b. \(B=\left\\{s_{2}, s_{4}, s_{5}, s_{6}\right\\}\) c. \(C=S\)

Let \(S\) be a sample space for an experiment, and let \(E\) and \(F\) be events of this experiment. Show that the events \(E \cup F\) and \(E^{c} \cap F^{c}\) are mutually exclusive. Hint: Use De Morgan's law.

In how many ways can a supermarket chain select 3 out of 12 possible sites for the construction of new supermarkets?

Let \(U\) denote the set of all employees in a hospital. Let $$ \begin{array}{l} \boldsymbol{N}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a nurse }\\} \\ \boldsymbol{D}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a doctor\\} } \\ \boldsymbol{A}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is an administrator\\} } \\ \boldsymbol{M}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a male\\} } \\ \boldsymbol{F}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a female }\\} \end{array} $$ Describe each set in words. a. \(D \cap M^{c}\) b. \(D \cap A\)

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}, s_{5}\right\\}\) be the sample space associated with an experiment having the following probability distribution: $$ \begin{array}{lccccc} \hline \text { Outcome } & s_{1} & s_{2} & s_{3} & s_{4} & s_{5} \\ \hline \text { Probability } & \frac{1}{14} & \frac{3}{14} & \frac{6}{14} & \frac{2}{14} & \frac{2}{14} \\ \hline \end{array} $$ Find the probability of the event: a. \(A=\left\\{s_{1}, s_{2}, s_{4}\right\\}\) b. \(B=\left\\{s_{1}, s_{5}\right\\}\) c. \(C=S\)

A student is given a reading list of ten books from which he must select two for an outside reading requirement. In how many ways can he make his selections?

Let \(U\) denote the set of all employees in a hospital. Let $$ \begin{array}{l} \boldsymbol{N}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a nurse }\\} \\ \boldsymbol{D}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a doctor\\} } \\ \boldsymbol{A}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is an administrator\\} } \\ \boldsymbol{M}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a male\\} } \\ \boldsymbol{F}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a female }\\} \end{array} $$ Describe each set in words. a. \(N \cap F\) b. \((D \cup N)^{c}\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. The function \(f(x)=2 e^{x}-\frac{1}{2}(\cos x+\sin x)\) is a solution of the differential equation \(y^{\prime}-y=\sin x\).

Suppose the probability that Bill can solve a problem is \(p_{1}\) and the probability that Mike can solve it is \(p_{2}\). Show that the probability that Bill and Mike working independently can solve the problem is \(p_{1}+p_{2}-p_{1} p_{2}\).

In how many ways can a quality-control engineer select a sample of 3 microprocessors for testing from a batch of 100 microprocessors?

Let \(U\) denote the set of all senators in Congress and let $$ \begin{array}{l} D=\\{x \in U \mid x \text { is a Democrat }\\} \\ R=\\{x \in U \mid x \text { is a Republican\\} } \\ F=\\{x \in U \mid x \text { is a female }\\} \\ L=\\{x \in U \mid x \text { is a lawyer }\\} \end{array} $$ Write the set that represents each statement. a. The set of all Democrats who are female b. The set of all Republicans who are male and are not lawyers

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or 7 ? Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The numbers 1,2, and 3 are written separately on three pieces of paper. These slips of paper are then placed in a bowl. If you draw two slips from the bowl, one at a time and without replacement, then the sample space for this experiment consists of six elements.

A group of five students studying for a bar exam has formed a study group. Each member of the group will be responsible for preparing a study outline for one of five courses. In how many different ways can the five courses be assigned to the members of the group?

Let \(U\) denote the set of all senators in Congress and let $$ \begin{array}{l} D=\\{x \in U \mid x \text { is a Democrat }\\} \\ R=\\{x \in U \mid x \text { is a Republican\\} } \\ F=\\{x \in U \mid x \text { is a female }\\} \\ L=\\{x \in U \mid x \text { is a lawyer }\\} \end{array} $$ Write the set that represents each statement. a. The set of all Democrats who are female or are lawyers b. The set of all senators who are not Democrats or are lawyers

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\) and \(P(B)=0\), then \(P(A)=0\).

In how many ways can a television-programming director schedule six different commercials in the six time slots allocated to commercials during a 1 -hr program?

Let \(U\) denote the set of all students in the business college of a certain university. Let \(A=\\{x \in U \mid x\) had taken a course in accounting \(B=\\{x \in U \mid x\) had taken a course in economics\\} \(C=\\{x \in U \mid x\) had taken a course in marketing \(\\}\) Write the set that represents each statement. a. The set of students who have not had a course in economics b. The set of students who have had courses in accounting and economics c. The set of students who have had courses in accounting and economics but not marketing

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\), then \(P(A) \leq P(B)\).

Seven people arrive at the ticket counter of a cinema at the same time. In how many ways can they line up to purchase their tickets?

Let \(U\) denote the set of all students in the business college of a certain university. Let \(A=\\{x \in U \mid x\) had taken a course in accounting \(B=\\{x \in U \mid x\) had taken a course in economics\\} \(C=\\{x \in U \mid x\) had taken a course in marketing \(\\}\) Write the set that represents each statement. a. The set of students who have had courses in economics but not courses in accounting or marketing b. The set of students who have had at least one of the three courses c. The set of students who have had all three courses

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(E\) is an event of an experiment, then \(P(E)+P\left(E^{c}\right)=1\).

A company car that has a seating capacity of six is to be used by six employees who have formed a car pool. If only four of these employees can drive, how many possible seating arrangements are there for the group?

At a college library exhibition of faculty publications, three mathematics books, four social science books, and three biology books will be displayed on a shelf. (Assume that none of the books is alike.) a. In how many ways can the ten books be arranged on the shelf? b. In how many ways can the ten books be arranged on the shelf if books on the same subject matter are placed together?

Use Venn diagrams to illustrate each statement. $$ A \subseteq A \cup B ; B \subseteq A \cup B \quad \text { 54. } A \cap B \subseteq A ; A \cap B \subseteq B $$

SEATING In how many ways can four married couples attending a concert be seated in a row of eight seats if a. There are no restrictions? b. Each married couple is seated together? c. The members of each sex are seated together?

Use Venn diagrams to illustrate each statement. $$ A \cup(B \cup C)=(A \cup B) \cup C $$

Use Venn diagrams to illustrate each statement. $$ A \cup(B \cup C)=(A \cup B) \cup C $$

\(C\) \& J Realty has received 12 inquiries from prospective home buyers. In how many ways can the inquiries be directed to any four of the firm's real estate agents if each agent handles three inquiries?

Use Venn diagrams to illustrate each statement. $$ A \cap(B \cap C)=(A \cap B) \cap C $$

A Little League baseball team has 12 players available for a 9-member team (no designated team positions). a. How many different 9 -person batting orders are possible? b. How many different 9-member teams are possible? c. How many different 9 -member teams and 2 alternates are possible?

Use Venn diagrams to illustrate each statement. $$ A \cap(B \cup C)=(A \cap B) \cup(A \cap C) $$

In the men's tennis tournament at Wimbledon, two finalists, \(\mathrm{A}\) and \(\mathrm{B}\), are competing for the title, which will be awarded to the first player to win three sets. In how many different ways can the match be completed?

Use Venn diagrams to illustrate each statement. $$ (A \cup B)^{c}=A^{c} \cap B^{c} $$

In the women's tennis tournament at Wimbledon, two finalists, \(\mathrm{A}\) and \(\mathrm{B}\), are competing for the title, which will be awarded to the first player to win two sets. In how many different ways can the match be completed?

Let $$ \begin{array}{l} U=\\{1,2,3,4,5,6,7,8,9,10\\} \\ A=\\{1,3,5,7,9\\} \\ B=\\{1,2,4,7,8\\} \\ C=\\{2,4,6,8\\} \end{array} $$ Verify each equation by direct computation. a. \(A \cup(B \cup C)=(A \cup B) \cup C\) b. \(A \cap(B \cap C)=(A \cap B) \cap C\)

In how many different ways can a panel of 12 jurors and 2 alternate jurors be chosen from a group of 30 prospective jurors?

Let $$ \begin{array}{l} U=\\{1,2,3,4,5,6,7,8,9,10\\} \\ A=\\{1,3,5,7,9\\} \\ B=\\{1,2,4,7,8\\} \\ C=\\{2,4,6,8\\} \end{array} $$ Verify each equation by direct computation. a. \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C)\) b. \((A \cup B)^{c}=A^{c} \cap B^{c}\)

A student taking an examination is required to answer exactly 10 out of 15 questions.

TEACHING AssISTANTSHIPS Twelve graduate students have applied for three available teaching assistantships. In how many ways can the assistantships be awarded among these applicants if a. No preference is given to any student? b. One particular student must be awarded an assistantship? c. The group of applicants includes seven men and five women and it is stipulated that at least one woman must be awarded an assistantship?

SENATE COMMITTEES In how many ways can a subcommittee of four be chosen from a Senate committee of five Democrats and four Republicans if a. All members are eligible? b. The subcommittee must consist of two Republicans and two Democrats?

Television Company is considering bids submitted by seven different firms for each of three different contracts. In how many ways can the contracts be awarded among these firms if no firm is to receive more than two contracts?

Suppose \(A \subset B\) and \(B \subset C\), where \(A\) and \(B\) are any two sets. What conclusion can be drawn regarding the sets \(A\) and \(C\) ?

Computers has five vacancies in its executive trainee program. In how many ways can the company select five trainees from a group of ten female and ten male applicants if the vacancies a. Can be filled by any combination of men and women? b. Must be filled by two men and three women?

Verify the assertion that two sets \(A\) and \(B\) are equal if and only if (1) \(A \subseteq B\) and (2) \(B \subseteq A\).

CoURSE SELECTION A student planning her curriculum for the upcoming year must select one of five business courses, one of three mathematics courses, two of six elective courses, and either one of four history courses or one of three social science courses. How many different curricula are available for her consideration?