Chapter 7

The manager of a local bank observes how long it takes a customer to complete his transactions at the automatic bank teller. a. Describe an appropriate sample space for this experiment. b. Describe the event that it takes a customer between 2 and 3 min to complete his transactions at the automatic bank teller.

In how many ways can five people line up at a checkout counter in a supermarket?

To help plan the number of meals (breakfast, lunch, and dinner) to be prepared in a college cafeteria, a survey was conducted and the following data were obtained: 130 students ate breakfast. 180 students ate lunch. 275 students ate dinner. 68 students ate breakfast and lunch. 112 students ate breakfast and dinner. 90 students ate lunch and dinner. 58 students ate all three meals. How many of the students ate a. At least one meal in the cafeteria? b. Exactly one meal in the cafeteria? c. Only dinner in the cafeteria? d. Exactly two meals in the cafeteria?

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) \(A=\\{1,3,5,7,9\\}, B=\\{2,4,6,8,10\\}\), and \(C=\\{1,2,4,\), \(5,8,9\\}\). List the elements of each set. a. \(C \cap C^{c}\) b. \((A \cap C)^{c}\) c. \(A \cup(B \cap C)\)

A survey in which people were asked how they were planning to prepare their taxes in 2007 revealed the following: $$ \begin{array}{lc} \hline \begin{array}{l} \text { Method of } \\ \text { Preparation } \end{array} & \text { Percent } \\ \hline \text { Computer software } & 33.9 \\ \hline \text { Accountant } & 23.6 \\ \hline \text { Tax preparation service } & 17.4 \\ \hline \text { Spouse, friend, or other } & \\ \text { relative will prepare } & 10.8 \\ \hline \text { By hand } & 14.3 \\ \hline \end{array} $$ What is the probability that a randomly chosen participant in the survey a. Was planning to use an accountant or a tax preparation service to prepare his taxes? b. Was not planning to use computer software to prepare his taxes and was not planning to do his taxes by hand?

Robin purchased shares of a machine tool company and shares of an airline company. Let \(E\) be the event that the shares of the machine tool company increase in value over the next \(6 \mathrm{mo}\), and let \(F\) be the event that the shares of the airline company increase in value over the next \(6 \mathrm{mo}\). Using the symbols \(\cup, \cap\), and \({ }^{c}\), describe the following events. a. The shares in the machine tool company do not increase in value. b. The shares in both the machine tool company and the airline company do not increase in value. c. The shares of at least one of the two companies increase in value. d. The shares of only one of the two companies increase in value.

How many different batting orders can be formed for a nine-member baseball team?

INVESTMENTS In a survey of 200 employees of a company regarding their \(401(\mathrm{k})\) investments, the following data were obtained: 141 had investments in stock funds. 91 had investments in bond funds. 60 had investments in money market funds. 47 had investments in stock funds and bond funds. 36 had investments in stock funds and money market funds. 36 had investments in bond funds and money market funds. 5 had investments only in some other vehicle. a. How many of the employees surveyed had investments in all three types of funds? b. How many of the employees had investments in stock funds only?

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) \(A=\\{1,3,5,7,9\\}, B=\\{2,4,6,8,10\\}\), and \(C=\\{1,2,4,\), \(5,8,9\\}\). List the elements of each set. a. \((A \cap B) \cup C\) b. \((A \cup B \cup C)^{c}\) c. \((A \cap B \cap C)^{c}\)

In an online survey for Talbots of 1095 women ages \(35 \mathrm{yr}\) and older, the participants were asked what article of clothing women most want to fit perfectly. A summary of the results of the survey follows: $$ \begin{array}{lc} \hline \text { Article of Clothing } & \text { Respondents } \\ \hline \text { Jeans } & 470 \\ \hline \text { Black Pantsuit } & 307 \\ \hline \text { Cocktail Dress } & 230 \\ \hline \text { White Shirt } & 22 \\ \hline \text { Gown } & 11 \\ \hline \text { Other } & 55 \\ \hline \end{array} $$ If a woman who participated in the survey is chosen at random, what is the probability that she most wants a. Jeans to fit perfectly? b. A black pantsuit or a cocktail dress to fit perfectly?

In a survey of 106 senior information technology and data security professionals at major U.S. companies regarding their confidence that they had detected all significant security breaches in the past year, the following responses were obtained. $$ \begin{array}{lcccc} \hline & \begin{array}{c} \text { Very } \\ \text { Answer } \end{array} & \begin{array}{c} \text { Moderately } \\ \text { confident } \end{array} & \begin{array}{c} \text { Not very } \\ \text { confident } \end{array} & \begin{array}{c} \text { Not at all } \\ \text { confident } \end{array} & \text { confident } \\ \hline \text { Respondents } & 21 & 56 & 22 & 7 \\ \hline \end{array} $$ What is the probability that a respondent in the survey selected at random a. Had little or no confidence that he or she had detected all significant security breaches in the past year? b. Was very confident that he or she had detected all significant security breaches in the past year?

The customer service department of Universal Instruments, manufacturer of the Galaxy home computer, conducted a survey among customers who had returned their purchase registration cards. Purchasers of its deluxe model home computer were asked to report the length of time \((t)\) in days before service was required. a. Describe a sample space corresponding to this survey. b. Describe the event \(E\) that a home computer required service before a period of 90 days had elapsed. c. Describe the event \(F\) that a home computer did not require service before a period of 1 yr had elapsed.

In how many ways can the names of six candidates for political office be listed on a ballot?

NEWSPAPER SUBSCRIPTIONS In a survey of 300 individual investors regarding subscriptions to the New York Times (NYT), Wall Street Journal (WSJ), and USA Today (UST), the following data were obtained: 122 subscribe to the \(N Y T\). 150 subscribe to the \(W S J\). 62 subscribe to the \(U S T\). 38 subscribe to the \(N Y T\) and \(W S J\). 20 subscribe to the \(N Y T\) and \(U S T\). 28 subscribe to the \(W S J\) and \(U S T\). 36 do not subscribe to any of these newspapers. a. How many of the individual investors surveyed subscribe to all three newspapers? b. How many subscribe to only one of these newspapers?

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) \(A=\\{1,3,5,7,9\\}, B=\\{2,4,6,8,10\\}\), and \(C=\\{1,2,4,\), \(5,8,9\\}\). List the elements of each set. a. \(A^{c} \cap\left(B \cap C^{c}\right)\) b. \(\left(A \cup B^{c}\right) \cup\left(B \cap C^{c}\right)\) c. \((A \cup B)^{c} \cap C^{c}\)

Two hundred workers were asked: Would a better economy lead you to switch jobs? The results of the survey follow: $$ \begin{array}{lccccc} \hline & \text { Very } & \text { Somewhat } & \text { Somewhat } & \text { Very } & \text { Don't } \\ \text { Answer } & \text { likely } & \text { likely } & \text { unlikely } & \text { unlikely } & \text { know } \\ \hline \text { Respondents } & 40 & 28 & 26 & 104 & 2 \\ \hline \end{array} $$ If a worker is chosen at random, what is the probability that he or she a. Is very unlikely to switch jobs? b. Is somewhat likely or very likely to switch jobs?

A time study was conducted by the production manager of Vista Vision to determine the length of time in minutes required by an assembly worker to complete a certain task during the assembly of its Pulsar color television sets. a. Describe a sample space corresponding to this time study. b. Describe the event \(E\) that an assembly worker took 2 min or less to complete the task. c. Describe the event \(F\) that an assembly worker took more than 2 min to complete the task.

In how many ways can a member of a hiring committee select 3 of 12 job applicants for further consideration?

Determine whether the pairs of sets are disjoint. a. \(\\{1,2,3,4\\},\\{4,5,6,7\\}\) b. \(\\{a, c, e, g\\},\\{b, d, f\\}\)

According to a study conducted in 2003 concerning the participation, by age, of \(401(\mathrm{k})\) investors, the following data were obtained: $$ \begin{array}{lccccc} \hline \text { Age } & 20 \mathrm{~s} & 30 \mathrm{~s} & 40 \mathrm{~s} & 50 \mathrm{~s} & 60 \mathrm{~s} \\ \hline \text { Percent } & 11 & 28 & 32 & 22 & 7 \\ \hline \end{array} $$ a. What is the probability that a \(401(\mathrm{k})\) investor selected at random is in his or her 20 s or 60 s? b. What is the probability that a \(401(\mathrm{k})\) investor selected at random is under the age of 50 ?

In a survey conducted in 2007 of 1402 workers 18 yr and older regarding their opinion on retirement benefits, the following data were obtained: 827 said that it was better to have excellent retirement benefits with a lower-than-expected salary, 477 said that it was better to have a higher-than-expected salary with poor retirement benefits, 42 said "neither," and 56 said "not sure." If a worker in the survey is selected at random, what is the probability that he or she answered that it was better to have a. Excellent retirement benefits with a lower-than-expected salary? b. A higher-than-expected salary with poor retirement benefits?

In how many ways can an investor select four mutual funds for his investment portfolio from a recommended list of eight mutual funds?

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 \(A \subseteq B\), then \(n(B)=n(A)+n\left(A^{c} \cap B\right)\).

Determine whether the pairs of sets are disjoint. a. \(\varnothing,\\{1,3,5\\}\) b. \(\\{0,1,3,4\\},\\{0,2,5,7\\}\)

In a poll conducted among likely voters by Zogby International, voters were asked their opinion on the best alternative to oil and coal. The results are as follows: $$ \begin{array}{lcccccc} \hline & & & \text { Fuel } & & \text { Other/ } \\ \text { Source } & \text { Nuclear } & \text { Wind } & \text { cells } & \text { Biofuels } & \text { Solar } & \text { no answer } \\ \hline \text { Respondents, } \% & 14.2 & 16.0 & 3.8 & 24.3 & 27.9 & 13.8 \\ \hline \end{array} $$ What is the probability that a randomly selected participant in the poll mentioned a. Wind or solar energy sources as the best alternative to oil and coal? b. Nuclear or biofuels as the best alternative to oil and coal?

In an attempt to study the leading causes of airline crashes, the following data were compiled from records of airline crashes from 1959 to 1994 (excluding sabotage and military action). $$ \begin{array}{lc} \hline \text { Primary Factor } & \text { Accidents } \\ \hline \text { Pilot } & 327 \\ \hline \text { Airplane } & 49 \\ \hline \text { Maintenance } & 14 \\ \hline \text { Weather } & 22 \\ \hline \text { Airport/air traffic control } & 19 \\ \hline \text { Miscellaneous/other } & 15 \\ \hline \end{array} $$ Assume that you have just learned of an airline crash and that the data give a generally good indication of the causes of airline crashes. Give an estimate of the probability that the primary cause of the crash was due to pilot error or bad weather.

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 \(n(A \cup B)=n(A)+n(B)\), then \(A \cap B=\varnothing\)

Let \(U\) denote the set of all employees at Universal Life Insurance Company and let $$ \begin{array}{l} \boldsymbol{T}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks tea }\\} \\ \boldsymbol{C}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks coffee }\\} \end{array} $$ Describe each set in words. a. \(T^{c}\) b. \(C^{c}\)

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?

In a survey conducted in the fall 2006, 800 homeowners were asked about their expectations regarding the value of their home in the next few years; the results of the survey are summarized below: $$ \begin{array}{lc} \hline \text { Expectations } & \text { Homeowners } \\ \hline \text { Decrease } & 48 \\ \hline \text { Stay the same } & 152 \\ \hline \text { Increase less than } 5 \% & 232 \\ \hline \text { Increase 5-10\% } & 240 \\ \hline \text { Increase more than 10\% } & 128 \\ \hline \end{array} $$ If a homeowner in the survey is chosen at random, what is the probability that he or she expected his or her home to a. Stay the same or decrease in value in the next few years? b. Increase \(5 \%\) or more in value in the next few years?

In the opinion poll of Exercise 38, the voters were also asked to indicate their political affiliations-Democrat, Republican, or Independent. As before, let the letters \(L, M\), and \(U\) represent the low-, middle-, and upper-income groups, respectively. Let the letters \(D, R\) and \(I\) represent Democrat, Republican, and Independent, respectively. a. Describe a sample space corresponding to this poll. b. Describe the event \(E_{1}\) that a respondent is a Democrat. c. Describe the event \(E_{2}\) that a respondent belongs to the upper-income group and is a Republican. d. Describe the event \(E_{3}\) that a respondent belongs to the middle-income group and is not a Democrat.

Find the number of distinguishable permutations that can be formed from the letters of the word PHILIPPINES.

Let \(U\) denote the set of all employees at Universal Life Insurance Company and let $$ \begin{array}{l} \boldsymbol{T}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks tea }\\} \\ \boldsymbol{C}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks coffee }\\} \end{array} $$ Describe each set in words. a. \(T \cup C\) b. \(T \cap C\)

The following table, compiled in 2004 , gives the percentage of music downloaded from the United States and other countries by U.S. users: $$ \begin{array}{lcccccccc} \hline \text { Country } & \text { U.S. } & \text { Germany } & \text { Canada } & \text { Italy } & \text { U.K. } & \text { France } & \text { Japan } & \text { Other } \\ \hline \text { Percent } & 45.1 & 16.5 & 6.9 & 6.1 & 4.2 & 3.8 & 2.5 & 14.9 \\\ \hline \end{array} $$ a. Verify that the table does give a probability distribution for the experiment. b. What is the probability that a user who downloads music, selected at random, obtained it from either the United States or Canada? c. What is the probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom (U.K.), or France?

A certain airport hotel operates a shuttle bus service between the hotel and the airport. The maximum capacity of a bus is 20 passengers. On alternate trips of the shuttle bus over a period of \(1 \mathrm{wk}\), the hotel manager kept a record of the number of passengers arriving at the hotel in each bus. a. What is an appropriate sample space for this experiment? b. Describe the event \(E\) that a shuttle bus carried fewer than ten passengers. c. Describe the event \(F\) that a shuttle bus arrived with a full load.

In how many ways can the letters of the Web site MySpace be arranged if all of the letters are used and the vowels \(a\) and \(e\) must always stay in the order \(a e\) ?

Derive Equation (5). Hint: Equation (4) can be written as \(n(D \cup E)=n(D)+n(E)-\) \(n(D \cap E) .\) Now, put \(D=A \cup B\) and \(E=C .\) Use (4) again if necessary.

Let \(U\) denote the set of all employees at Universal Life Insurance Company and let $$ \begin{array}{l} \boldsymbol{T}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks tea }\\} \\ \boldsymbol{C}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks coffee }\\} \end{array} $$ Describe each set in words. a. \(T \cap C^{c}\) b. \(T^{c} \cap C\)

In a survey conducted to see how long Americans keep their cars, 2000 automobile owners were asked how long they plan to keep their present cars. The results of the survey follow: $$ \begin{array}{cc} \hline \text { Years Car Is Kept, } \boldsymbol{x} & \text { Respondents } \\ \hline 0 \leq x<1 & 60 \\ \hline 1 \leq x<3 & 440 \\ \hline 3 \leq x<5 & 360 \\ \hline 5 \leq x<7 & 340 \\ \hline 7 \leq x<10 & 240 \\ \hline 10 \leq x & 560 \\ \hline \end{array} $$ Find the probability distribution associated with these data. What is the probability that an automobile owner selected at random from those surveyed plans to keep his or her present car a. Less than \(5 \mathrm{yr}\) ? b. 3 yr or more?

Determine whether the given experiment has a sample space with equally likely outcomes. A loaded die is rolled, and the number appearing uppermost on the die is recorded.

Eight players, \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}\), and \(\mathrm{H}\), are com- peting in a series of elimination matches of a tennis tournament in which the winner of each preliminary match will advance to the semifinals and the winners of the semifinals will advance to the finals. An outline of the scheduled matches follows. Describe a sample space listing the possible participants in the finals.

In how many ways can five people boarding a bus be seated if the bus has eight vacant seats?

Let \(U\) denote the set of all employees at Universal Life Insurance Company and let $$ \begin{array}{l} \boldsymbol{T}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks tea }\\} \\ \boldsymbol{C}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks coffee }\\} \end{array} $$ Describe each set in words. a. \(T^{c} \cap C^{c}\) b. \((T \cup C)^{c}\)

A time study was conducted by the production manager of Universal Instruments to determine how much time it took an assembly worker to complete a certain task during the assembly of its Galaxy home computers. Results of the study indicated that \(20 \%\) of the workers were able to complete the task in less than \(3 \mathrm{~min}, 60 \%\) of the workers were able to complete the task in 4 min or less, and \(10 \%\) of the workers required more than 5 min to complete the task. If an assembly-line worker is selected at random from this group, what is the probability that a. He or she will be able to complete the task in 5 min or less? b. He or she will not be able to complete the task within 4 min? c. The time taken for the worker to complete the task will be between 3 and 4 min (inclusive)?

Determine whether the given experiment has a sample space with equally likely outcomes. Two fair dice are rolled, and the sum of the numbers appearing uppermost is recorded.

An experiment consists of selecting a card at random from a well-shuffled 52 -card deck. Let \(E\) denote the event that an ace is drawn and let \(F\) denote the event that a spade is drawn. Show that \(n(E \cup F)=n(E)+n(F)-n(E \cap F)\).

How many distinct five-digit numbers can be made using the digits \(1,2,2,2,7 ?\)

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. \(D^{c}\) b. \(N^{c}\)

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) \(\begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside car } \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned}\) 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?

Determine whether the given experiment has a sample space with equally likely outcomes. A ball is selected at random from an urn containing six black balls and six red balls, and the color of the ball is recorded.