Chapter 7

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets. $$ \begin{array}{l} \text { \\{GM, Ford, Chrysler\\}, \\{Daimler-Benz, Volkswagen\\}, \\{Toy- }\\\ \text { ota, Nissan\\} } \end{array} $$

Let \(E\) and \(F\) be two events that are mutually exclusive, and suppose \(P(E)=.2\) and \(P(F)=.5\). Compute: a. \(P(E \cap F)\) b. \(P(E \cup F)\) c. \(P\left(E^{c}\right)\) d. \(P\left(E^{c} \cap F^{c}\right)\)

If a card is drawn at random from a standard 52 -card deck, what is the probability that the card drawn is a. A diamond? b. A black card? c. An ace?

An experiment consists of selecting a card from a standard deck of playing cards and noting whether it is black \((B)\) or red \((R)\). a. Describe an appropriate sample space for this experiment. b. What are the events of this experiment?

Classify each problem according to whether it involves a permutation or a combination. As part of a quality-control program, 3 cell phones are selected at random for testing from 100 cell phones produced by the manufacturer. In how many ways can this test batch be chosen?

SLOT MACHINES A "lucky dollar" is one of the nine symbols printed on each reel of a slot machine with three reels. A player receives one of various payouts whenever one or more "lucky dollars" appear in the window of the machine. Find the number of winning combinations for which the machine gives a payoff. Hint: (a) Compute the number of ways in which the nine symbols on the first, second, and third wheels can appear in the window slot and (b) compute the number of ways in which the eight symbols other than the "lucky dollar" can appear in the window slot. The difference \((a-b)\) is the number of ways in which the "lucky dollar" can appear in the window slot. Why?

Let \(A, B\), and \(C\) be subsets of a universal set \(U\) and suppose \(n(U)=100, n(A)=28, n(B)=30\) \(n(C)=34, n(A \cap B)=8, n(A \cap C)=10, n(B \cap C)=15\) and \(n(A \cap B \cap C)=5\). Compute: a. \(n(A \cup B \cup C)\) b. \(n\left(A^{c} \cap B \cap C\right)\)

Use Venn diagrams to represent the following relationships: a. \(A \subset B\) and \(B \subset C\) b. \(A \subset U\) and \(B \subset U\), where \(A\) and \(B\) have no elements in common c. The sets \(A, B\), and \(C\) are equal.

Let \(E\) and \(F\) be two events of an experiment with sample space \(S\). Suppose \(P(E)=.6, P(F)=.4\), and \(P(E \cap F)=\) .2. Compute: a. \(P(E \cup F)\) b. \(P\left(E^{c}\right)\) c. \(P\left(F^{c}\right)\) d. \(P\left(E^{c} \cap F\right)\)

A pair of fair dice is rolled. What is the probability that a. The sum of the numbers shown uppermost is less than 5 ? b. At least one 6 is rolled?

An experiment consists of selecting a letter at random from the letters in the word MASSACHUSETTS and observing the outcomes. a. What is an appropriate sample space for this experiment? b. Describe the event "the letter selected is a vowel."

Classify each problem according to whether it involves a permutation or a combination. How many three-digit numbers can be formed using the numerals in the set \(\\{3,2,7,9\\}\) if repetition is not allowed?

STAFFING Student Painters, which specializes in painting the exterior of residential buildings, has five people available to be organized into two- person and three-person teams. a. In how many ways can a two-person team be formed? b. In how many ways can a three-person team be formed? c. In how many ways can the company organize the available people into either two-person teams or threeperson teams?

Let \(A, B\), and \(C\) be subsets of a universal set \(U\) and suppose \(n(U)=100, n(A)=28, n(B)=30\) \(n(C)=34, n(A \cap B)=8, n(A \cap C)=10, n(B \cap C)=15\) and \(n(A \cap B \cap C)=5\). Compute: a. \(n[A \cap(B \cup C)]\) b. \(n\left[A \cap(B \cup C)^{c}\right]\)

Let \(U\) denote the set of all students who applied for admission to the freshman class at Faber College for the upcoming academic year, and let \(A=\\{x \in U \mid x\) is a successful applicant \(\\}\) \(B=\\{x \in U \mid x\) is a female student who enrolled in the freshman class\\} \(C=\\{x \in U \mid x\) is a male student who enrolled in the freshman class\\} a. Use Venn diagrams to represent the sets \(U, A, B\), and \(C\). b. Determine whether the following statements are true or false. i. \(A \subseteq B\) ii. \(B \subset A\) iii. \(C \subset B\)

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}\right\\}\) be the sample space associated with an experiment having the probability distribution shown in the accompanying table. If \(A=\left\\{s_{1}, s_{2}\right\\}\) and \(B=\left\\{s_{1}, s_{3}\right\\}\), find a. \(P(A), P(B)\) b. \(P\left(A^{c}\right), P\left(B^{c}\right)\) c. \(P(A \cap B)\) d. \(P(A \cup B)\) $$ \begin{array}{lc} \hline \text { Outcome } & \text { Probability } \\ \hline s_{1} & \frac{1}{8} \\ \hline s_{2} & \frac{3}{8} \\ \hline s_{3} & \frac{1}{4} \\ \hline s_{4} & \frac{1}{4} \\ \hline \end{array} $$

What is the probability of arriving at a traffic light when it is red if the red signal is lit for \(30 \mathrm{sec}\), the yellow signal for \(5 \mathrm{sec}\), and the green signal for \(45 \mathrm{sec}\) ?

An experiment consists of tossing a coin, rolling a die, and observing the outcomes. a. Describe an appropriate sample space for this experiment. b. Describe the event "a head is tossed and an even number is rolled."

Classify each problem according to whether it involves a permutation or a combination. In how many ways can nine different books be arranged on a shelf?

Let \(A, B\), and \(C\) be subsets of a universal set \(U\) and suppose \(n(U)=100, n(A)=28, n(B)=30\) \(n(C)=34, n(A \cap B)=8, n(A \cap C)=10, n(B \cap C)=15\) and \(n(A \cap B \cap C)=5\). Compute: a. \(n\left(A^{c} \cap B^{c} \cap C^{c}\right)\) b. \(n\left[A^{c} \cap(B \cup C)\right]\)

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 probability distribution shown in the accompanying table. If \(A=\left\\{s_{1}, s_{2}\right\\}\) and \(B=\left\\{s_{1}, s_{5}, s_{6}\right\\}\), find a. \(P(A), P(B)\) b. \(P\left(A^{c}\right), P\left(B^{c}\right)\) c. \(P(A \cap B)\) d. \(P(A \cup B)\) e. \(P\left(A^{c} \cap B^{c}\right)\) f. \(P\left(A^{c} \cup B^{c}\right)\) $$ \begin{array}{cc} \hline \text { Outcome } & \text { Probability } \\ \hline s_{1} & \frac{1}{3} \\ \hline s_{2} & \frac{1}{8} \\ \hline s_{3} & \frac{1}{6} \\ \hline s_{4} & \frac{1}{6} \\ \hline s_{5} & \frac{1}{12} \\ \hline s_{6} & \frac{1}{8} \\ \hline \end{array} $$

What is the probability that a roulette ball will come to rest on an even number other than 0 or 00 ? (Assume that there are 38 equally likely outcomes consisting of the numbers \(1-36,0\), and 00 .)

Classify each problem according to whether it involves a permutation or a combination. A member of a book club wishes to purchase two books from a selection of eight books recommended for a certain month. In how many ways can she choose them?

A nonprofit organization conducted a survey of 2140 metropolitan-area teachers regarding their beliefs about educational problems. The following data were obtained: 900 said that lack of parental support is a problem. 890 said that abused or neglected children are problems. 680 said that malnutrition or students in poor health is a problem. 120 said that lack of parental support and abused or neglected children are problems. 110 said that lack of parental support and malnutrition or poor health are problems. 140 said that abused or neglected children and malnutrition or poor health are problems. 40 said that lack of parental support, abuse or neglect, and malnutrition or poor health are problems. What is the probability that a teacher selected at random from this group said that lack of parental support is the only problem hampering a student's schooling? Hint: Draw a Venn diagram.

DISPOSITION OF CRIMINAL CASES Of the 98 first-degree murder cases from 2002 through the first half of 2004 in the Suffolk superior court, 9 cases were thrown out of the system, 62 cases were plea-bargained, and 27 cases went to trial. What is the probability that a case selected at random a. Was settled through plea bargaining? b. Went to trial?

A die is rolled and the number that falls uppermost is observed. Let \(E\) denote the event that the number shown is a 2 , and let \(F\) denote the event that the number shown is an even number. a. Are the events \(E\) and \(F\) mutually exclusive? b. Are the events \(E\) and \(F\) complementary?

A survey of the opinions of 10 leading economists in a certain country showed that, because oil prices were expected to drop in that country over the next 12 months, 7 had lowered their estimate of the consumer inflation rate. 8 had raised their estimate of the gross national product (GNP) growth rate. 2 had lowered their estimate of the consumer inflation rate but had not raised their estimate of the GNP growth rate. How many economists had both lowered their estimate of the consumer inflation rate and raised their estimate of the GNP growth rate for that period?

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. 22 had investments in stock funds, bond funds, and money market funds. What is the probability that an employee of the company chosen at random a. Had investments in exactly two kinds of investment funds? b. Had investments in exactly one kind of investment fund? c. Had no investment in any of the three types of funds?

In a sweepstakes sponsored by Gemini Paper Products, 100,000 entries have been received. If 1 grand prize, 5 first prizes, 25 second prizes, and 500 third prizes are to be awarded, what is the probability that a person who has submitted one entry will win a. The grand prize? b. A prize?

A die is rolled and the number that falls uppermost is observed. Let \(E\) denote the event that the number shown is even, and let \(F\) denote the event that the number is an odd number. a. Are the events \(E\) and \(F\) mutually exclusive? b. Are the events \(E\) and \(F\) complementary?

Classify each problem according to whether it involves a permutation or a combination. In how many ways can a six-letter security password be formed from letters of the alphabet if no letter is repeated?

STUDENT DROPOUT RATE Data released by the Department of Education regarding the rate (percentage) of ninth-grade students who don't graduate showed that, out of 50 states, 12 states had an increase in the dropout rate during the past 2 yr. 15 states had a dropout rate of at least \(30 \%\) during the past 2 yr. 21 states had an increase in the dropout rate and/or a dropout rate of at least \(30 \%\) during the past 2 yr. a. How many states had both a dropout rate of at least \(30 \%\) and an increase in the dropout rate over the 2 -yr period? b. How many states had a dropout rate that was less than \(30 \%\) but that had increased over the 2 -yr period?

The probability that a shopper in a certain boutique will buy a blouse is .35, that she will buy a pair of pants is \(.30\) and that she will buy a skirt is \(.27\). The probability that she will buy both a blouse and a skirt is \(.15\), that she will buy both a skirt and a pair of pants is \(.19\), and that she will buy both a blouse and a pair of pants is \(.12\). Finally, the probability that she will buy all three items is .08. What is the probability that a customer will buy a. Exactly one of these items? b. None of these items?

An opinion poll was conducted among a group of registered voters in a certain state concerning a proposition aimed at limiting state and local taxes. Results of the poll indicated that \(35 \%\) of the voters favored the proposition, \(32 \%\) were against it, and the remaining group were undecided. If the results of the poll are assumed to be representative of the opinions of the state's electorate, what is the probability that a registered voter selected at random from the electorate a. Favors the proposition? b. Is undecided about the proposition?

A sample of three transistors taken from a local electronics store was examined to determine whether the transistors were defective \((d)\) or nondefective \((n) .\) What is an appropriate sample space for this experiment?

How many four-letter permutations can be formed from the first four letters of the alphabet?

A survey of 100 college students who frequent the reading lounge of a university revealed the following results: 40 read Time. 30 read Newsweek. 25 read U.S. News \& World Report. 15 read Time and Newsweek. 12 read Time and U.S. News \& World Report. 10 read Newsweek and U.S. News \& World Report. 4 read all three magazines. How many of the students surveyed read a. At least one of these magazines? b. Exactly one of these magazines? c. Exactly two of these magazines? d. None of these magazines?

CoURSE ENROLLMENTS Among 500 freshmen pursuing a business degree at a university, 320 are enrolled in an economics course, 225 are enrolled in a mathematics course, and 140 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in a. An economics and/or a mathematics course? b. Exactly one of these two courses? c. Neither an economics course nor a mathematics course?

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), \(\mathrm{B}\) (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)?

Human blood is classified by the presence or absence of three main antigens (A, B, and Rh). When a blood specimen is typed, the presence of the \(\mathrm{A}\) and/or \(\mathrm{B}\) antigen is indicated by listing the letter \(A\) and/or the letter \(B\). If neither the A nor B antigen is present, the letter \(\mathrm{O}\) is used. The presence or absence of the \(\mathrm{Rh}\) antigen is indicated by the symbols \(+\) or \(-\), respectively. Thus, if a blood specimen is classified as \(\mathrm{AB}^{+}\), it contains the \(\mathrm{A}\) and the \(\mathrm{B}\) antigens as well as the \(\mathrm{Rh}\) antigen. Similarly, \(\mathrm{O}^{-}\) blood contains none of the three antigens. Using this information, determine the sample space corresponding to the different blood groups.

How many three-letter permutations can be formed from the first five letters of the alphabet?

SAT ScORES Results of a Department of Education survey of SAT test scores in 22 states showed that 10 states had an average composite SAT score of at least 1000 during the past 3 yr. 15 states had an increase of at least 10 points in the average composite SAT score during the past 3 yr. 8 states had both an average composite SAT score of at least 1000 and an increase in the average composite SAT score of at least 10 points during the past 3 yr. a. How many of the 22 states had composite SAT scores of less than 1000 and showed an increase of at least 10 points over the 3 -yr period? b. How many of the 22 states had composite SAT scores of at least 1000 and did not show an increase of at least 10 points over the 3 -yr period?

A leading manufacturer of kitchen appliances advertised its products in two magazines: Good Housekeeping and the Ladies Home Journal. A survey of 500 customers revealed that 140 learned of its products from Good Housekeeping, 130 learned of its products from the Ladies Home Journal, and 80 learned of its products from both magazines. What is the probability that a person selected at random from this group saw the manufacturer's advertisement in a. Both magazines? b. At least one of the two magazines? c. Exactly one magazine?

In a survey conducted in 2007 of 1004 adults 18 yr and older, the following question was asked: How are American companies doing on protecting the environment compared with companies in other countries? The results are summarized below: $$ \begin{array}{lcccc} \hline \text { Answer } & \text { Behind } & \text { Equal } & \text { Ahead } & \text { Don't know } \\ \hline \text { Respondents } & 382 & 281 & 251 & 90 \\ \hline \end{array} $$ If an adult in the survey is selected at random, what is the probability that he or she said that American companies are equal or ahead on protecting the environment compared with companies in other countries?

In a television game show, the winner is asked to select three prizes from five different prizes, \(A, B\), \(\mathrm{C}, \mathrm{D}\), and \(\mathrm{E} .\) a. Describe a sample space of possible outcomes (order is not important). b. How many points are there in the sample space corresponding to a selection that includes A? c. How many points are there in the sample space corresponding to a selection that includes \(\mathrm{A}\) and \(\mathrm{B}\) ? d. How many points are there in the sample space corresponding to a selection that includes either \(\mathrm{A}\) or \(\mathrm{B}\) ?

In how many ways can four students be seated in a row of four seats?

The 120 consumers of Exercise 19 were also asked about their buying preferences concerning another product that is sold in the market under three labels. The results were 12 buy only those sold under label A. 25 buy only those sold under label B. 26 buy only those sold under label C. 15 buy only those sold under labels \(\mathrm{A}\) and \(\mathrm{B}\). 10 buy only those sold under labels \(\mathrm{A}\) and \(\mathrm{C}\). 12 buy only those sold under labels \(\mathrm{B}\) and \(\mathrm{C}\). 8 buy the product sold under all three labels. How many of the consumers surveyed buy the product sold under a. At least one of the three labels? b. Labels A and B but not C? c. Label \(\mathrm{A}\) ? d. None of these labels?

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. \(A^{c}\) b. \(B \cup C\) c. \(C \cup C^{c}\)

The following table gives the number of people killed in rollover crashes in various types of vehicles in 2002 : Find the empirical probability distribution associated with these data. If a fatality due to a rollover crash in 2002 is picked at random, what is the probability that the victim was in a. \(\mathrm{A}\) car? b. An SUV? c. A pickup or an SUV?

STAYING IN ToucH In a poll conducted in 2007, 2000 adults ages 18 yr and older were asked how frequently they are in touch with their parents by phone. The results of the poll are as follows: $$ \begin{array}{lccccc} \hline \text { Answer } & \text { Monthly } & \text { Weekly } & \text { Daily } & \text { Don't know } & \text { Less } \\ \hline \text { Respondents, \% } & 11 & 47 & 32 & 2 & 8 \\ \hline \end{array} $$ If a person who participated in the poll is selected at random, what is the probability that the person said he or she kept in touch with his or her parents a. Once a week? b. At least once a week?