Chapter 7

In a survey conducted by a union, members were asked to rate the importance of the following issues: (1) job security, (2) increased fringe benefits, and (3) improved working conditions. Five different responses were allowed for each issue. Among completed surveys, how many different responses to this survey were possible?

If \(n(A)=10, n(A \cup B)=15\), and \(n(B)=8\), then what is \(n(A \cap B) ?\)

List the elements of the set in roster notation. $$ \\{x \mid 2-x=4 \text { and } x \text { is a fraction }\\} $$

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A face card (i.e., a jack, queen, or king) is drawn.

The grade distribution for a certain class is shown in the following table. Find the probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \\ \hline \text { Frequency of } & & & & & \\ \text { Occurrence } & 4 & 10 & 18 & 6 & 2 \\ \hline \end{array} $$

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Find the event \((E \cup F \cup G)^{c}\).

Evaluate the given expression. $$ P(n, 1) $$

A new state employee is offered a choice of ten basic health plans, three dental plans, and two vision care plans. How many different health-care plans are there to choose from if one plan is selected from each category?

Let \(A\) and \(B\) be subsets of a universal set \(U\) and suppose \(n(U)=200, n(A)=100, n(B)=80\), and \(n(A \cap B)=40\). Compute: a. \(n(A \cup B)\) b. \(n\left(A^{c}\right)\) c. \(n\left(A \cap B^{c}\right)\)

State whether the statements are true or false. a. \(\\{a, b, c\\}=\\{c, a, b\\}\) b. \(A \in A\)

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A red face card is drawn.

The percentage of the general population that has each blood type is shown in the following table. Determine the probability distribution associated with these data. $$ \begin{array}{lcccc} \hline \text { Blood Type } & \text { A } & \text { B } & \text { AB } & \text { O } \\ \hline \text { Population, \% } & 41 & 12 & 3 & 44 \\ \hline \end{array} $$

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Find the event \((E \cap F \cap G)^{c}\).

How many three-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions are allowed?

Let \(A\) and \(B\) be subsets of a universal set \(U\) and suppose \(n(U)=200, n(A)=100, n(B)=80\), and \(n(A \cap B)=40\). Compute: a. \(n\left(A^{c} \cap B\right)\) b. \(n\left(B^{c}\right)\). c. \(n\left(A^{c} \cap B^{c}\right)\)

State whether the statements are true or false. a. \(\varnothing \in A\) b. \(A \subset A\)

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A red face card is drawn.An ace is not drawn.

In a survey of 2000 adults 18 yr and older conducted in 2007 , the following question was asked: Is your family income keeping pace with the cost of living? The results of the survey follow: $$ \begin{array}{lcccc} \hline & \begin{array}{c} \text { Falling } \\ \text { behind } \end{array} & \begin{array}{c} \text { Staying } \\ \text { even } \end{array} & \begin{array}{c} \text { Increasing } \\ \text { faster } \end{array} & \begin{array}{c} \text { Don't } \\ \text { know } \end{array} \\ \hline \text { Respondents } & 800 & 880 & 240 & 80 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Evaluate the given expression. $$ C(6,6) $$

A Social Security number has nine digits. How many Social Security numbers are possible?

Find \(n(A \cup B)\) given that \(n(A)=6, n(B)=10\), and \(n(A \cap B)=3 .\)

State whether the statements are true or false. a. \(0 \in \varnothing\) b. \(0=\varnothing\)

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A black face card is not drawn.

In an online survey of 500 adults living with children under the age of \(18 \mathrm{yr}\), the participants were asked how many days per week they cook at home. The results of the survey are summarized below: $$ \begin{array}{lcccccccc} \hline \text { Number of Days } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \text { Respondents } & 25 & 30 & 45 & 75 & 55 & 100 & 85 & 85 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Are the events \(F\) and \(G\) mutually exclusive?

Evaluate the given expression. $$ C(8,8) $$

Computers manufactured by a certain company have a serial number consisting of a letter of the alphabet followed by a four-digit number. If all the serial numbers of this type have been used, how many sets have already been manufactured?

Five hundred raffle tickets were sold. What is the probability that a person holding one ticket will win the first prize? What is the probability that he or she will not win the first prize?

In a Los Angeles Times poll of 1936 California residents conducted in February 2004 , the following question was asked: Do you favor or oppose an amendment to the U.S. Constitution barring same-sex marriage? The following results were obtained: $$ \begin{array}{lccc} \hline \text { Opinion } & \text { Favor } & \text { Oppose } & \text { Don't know } \\ \hline \text { Respondents } & 910 & 891 & 135 \\ \hline \end{array} $$ Determine the empirical probability distribution associated with these data.

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Are the events \(E\) and \(F\) complementary?

Evaluate the given expression. $$ C(7,4) $$

A computer dating service uses the results of its compatibility survey for arranging dates. The survey consists of 50 questions, each having five possible answers. How many different responses are possible if every question is answered?

If \(n(A)=4, n(B)=5\), and \(n(A \cup B)=9\), find \(n(A \cap B)\).

State whether the statements are true or false. $$ \begin{array}{l} \text { \\{Chevrolet, Pontiac, Buick }\\} \subset\\{x \mid x \text { is a division of General }\\\ \text { Motors\\} } \end{array} $$

The results of a recent television survey of American TV households revealed that 87 out of every 100 TV households have at least one remote control. What is the probability that a randomly selected TV household does not have at least one remote control?

A study of deaths in car crashes from 1986 to 2002 revealed the following data on deaths in crashes by day of the week. $$ \begin{array}{lcccc} \hline \text { Day of the Week } & \text { Sunday } & \text { Monday } & \text { Tuesday } & \text { Wednesday } \\ \hline \begin{array}{l} \text { Average Number } \\ \text { of Deaths } \end{array} & 132 & 98 & 95 & 98 \\ \hline \text { Day of the Week } & \text { Thursday } & \text { Friday } & \text { Saturday } & \\ \hline \text { Average Number } & & & & \\ \text { of Deaths } & 105 & 133 & 158 & \\ \hline \end{array} $$ Find the empirical probability distribution associated with these data.

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Are the events \(F\) and \(G\) complementary?

Evaluate the given expression. $$ C(9,3) $$

The 2007 BMW 335i Coupe is offered with a choice of 14 exterior colors (11 metallic and 3 standard), 5 interior colors, and 4 trims. How many combinations involving color and trim are available for the model?

If \(n(A)=16, n(B)=16, n(C)=14, n(A \cap B)=6\), \(n(A \cap C)=5, n(B \cap C)=6\), and \(n(A \cup B \cup C)=31\), find \(n(A \cap B \cap C) .\)

In a poll conducted among 2000 college freshmen to ascertain the political views of college students. the accompanying data were obtained. Determine the empirical probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Political Views } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 52 & 398 & 1140 & 386 & 24 \\ \hline \end{array} $$ A: Far left B: Liberal C: Middle of the road D: Conservative E: Far right

Let \(S\) be any sample space and let E, \(\boldsymbol{F}\), and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(\cup, \cap\), and . The event that \(E\) and/or \(F\) occurs

Evaluate the given expression. $$ C(5,0) $$

The 2007 Toyota Camry comes with 5 grades of models, 2 sizes of engines, 4 choices of transmissions, 5 exterior colors, and 2 interior colors. How many choices of the Camry are available for a prospective buyer?

If \(n(A)=12, n(B)=12, n(A \cap B)=5, n(A \cap C)=5\), \(n(B \cap C)=4, n(A \cap B \cap C)=2\), and \(n(A \cup B \cup C)=\) 25, find \(n(C)\).

Let \(A=\\{1,2,3,4,5\\} .\) Determine whether the statements are true or false. a. \(2 \in A\) b. \(A \subseteq\\{2,4,6\\}\)

The accompanying data were obtained from a survey of 1500 Americans who were asked: How safe are American-made consumer products? Determine the empirical probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Rating } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 285 & 915 & 225 & 30 & 45 \\ \hline \end{array} $$ A: Very safe B: Somewhat safe C: Not too safe D: Not safe at all E: Don't know

Let \(S\) be any sample space and let E, \(\boldsymbol{F}\), and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(\cup, \cap\), and . The event that both \(E\) and \(F\) occur

Evaluate the given expression. $$ C(6,5) $$

An opinion poll is to be conducted among cable TV viewers. Six multiple-choice questions, each with four possible answers, will be asked. In how many different ways can a viewer complete the poll if exactly one response is given to each question?