Chapter 7

A survey of 1000 subscribers to the Los Angeles Times revealed that 900 people subscribe to the daily morning edition and 500 subscribe to both the daily morning and the Sunday editions. How many subscribe to the Sunday edition? How many subscribe to the Sunday edition only?

Let \(A=\\{1,2,3,4,5\\} .\) Determine whether the statements are true or false. a. \(0 \in A\) b. \(\\{1,3,5\\} \in A\)

The number of cars entering a tunnel leading to an airport in a major city over a period of 200 peak hours was observed, and the following data were obtained: $$ \begin{array}{rc} \hline \begin{array}{l} \text { Number of } \\ \text { Cars, } x \end{array} & \begin{array}{c} \text { Frequency of } \\ \text { Occurrence } \end{array} \\ \hline 01000 & 15 \\ \hline \end{array} $$ a. Describe an appropriate sample space for this experiment. b. Find the empirical probability distribution for this experiment.

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 \(G\) does not occur

Evaluate the given expression. $$ C(9,6) $$

To gain access to his account, a customer using an automatic teller machine (ATM) must enter a four-digit code. If repetition of the same four digits is not allowed (for example, 5555 ), how many possible combinations are there?

On a certain day, the Wilton County Jail held 190 prisoners accused of a crime (felony and/or misdemeanor). Of these, 130 were accused of felonies and 121 were accused of misdemeanors. How many prisoners were accused of both a felony and a misdemeanor?

Let \(A=\\{1,2,3\\}\). Which of the following sets are equal to \(A\) ? a. \(\\{2,1,3\\}\) b. \(\\{3,2,1\\}\) c. \(\\{0,1,2,3\\}\)

The arrival times of the 8 a.m. Bostonbased commuter train as observed in the suburban town of Sharon over 120 weekdays is summarized below: $$ \begin{array}{lc} \hline & \begin{array}{c} \text { Frequency of } \\ \text { Arrival Time, } \boldsymbol{x} \end{array} & \text { Occurrence } \\ \hline 7: 56 \text { a.m. }

Explain why the statement is incorrect. The probability that a certain stock will increase in value over a period of 1 week is .6. Therefore, the probability that the stock will decrease in value is \(.4\)

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 \(E\) but not \(F\) occurs

Evaluate the given expression. $$ C(10,3) $$

An opinion poll was conducted by the Morris Polling Group. Respondents were classified according to their sex (M or F), political affiliation (D, I, R), and the region of the country in which they reside (NW, W, C, \(\mathrm{S}, \mathrm{E}, \mathrm{NE})\) a. Use the generalized multiplication principle to determine the number of possible classifications. b. Construct a tree diagram to exhibit all possible classifications of females.

Of 100 clock radios with digital tuners and/or CD players sold recently in a department store, 70 had digital tuners and 90 had CD players. How many radios had both digital tuners and CD players?

Let \(A=\\{a, e, l, t, r\\} .\) Which of the following sets are equal to \(A\) ? a. \(\\{x \mid x\) is a letter of the word later\\} b. \(\\{x \mid x\) is a letter of the word latter \(\\}\) c. \(\\{x \mid x\) is a letter of the word relate \(\\}\)

Explain why the statement is incorrect. A red die and a green die are tossed. The probability that a 6 will appear uppermost on the red die is \(\frac{1}{6}\), and the probability that a 1 will appear uppermost on the green die is \(\frac{1}{6}\). Hence, the probability that the red die will show a 6 or the green die will show a 1 is \(\frac{1}{6}+\frac{1}{6}\).

According to Mediamark Research, 84 million out of 179 million adults in the United States correct their vision by using prescription eyeglasses, bifocals, or contact lenses. (Some respondents use more than one type.) What is the probability that an adult selected at random from the adult population uses corrective lenses?

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 \(E\) occurs but neither of the events \(F\) or \(G\) occurs

Evaluate the given expression. $$ C(n, 2) $$

Over the years, the state of California has used different combinations of letters of the alphabet and digits on its automobile license plates. a. At one time, license plates were issued that consisted of three letters followed by three digits. How many different license plates can be issued under this arrangement? b. Later on, license plates were issued that consisted of three digits followed by three letters. How many different license plates can be issued under this arrangement?

In a survey of 120 consumers conducted in a shopping mall, 80 consumers indicated that they buy brand A of a certain product, 68 buy brand \(\mathrm{B}\), and 42 buy both brands. How many consumers participating in the survey buy a. At least one of these brands? b. Exactly one of these brands? c. Only brand \(\mathrm{A}\) ? d. None of these brands?

List all subsets of the following sets: a. \(\\{1,2\\}\) b. \(\\{1,2,3\\}\) c. \(\\{1,2,3,4\\}\)

A study conducted by the Corrections Department of a certain state revealed that 163,605 people out of a total adult population of \(1,778,314\) were under correctional supervision (on probation, on parole, or in jail). What is the probability that a person selected at random from the adult population in that state is under correctional supervision?

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 \(E\) occurs but neither of the events \(F\) or \(G\) occurs

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

In recent years, the state of California issued license plates using a combination of one letter of the alphabet followed by three digits, followed by another three letters of the alphabet. How many different license plates can be issued using this configuration?

In a survey of 200 members of a local sports club, 100 members indicated that they plan to attend the next Summer Olympic Games, 60 indicated that they plan to attend the next Winter Olympic Games, and 40 indicated that they plan to attend both games. How many members of the club plan to attend a. At least one of the two games? b. Exactly one of the games? c. The Summer Olympic Games only? d. None of the games?

List all subsets of the set \(A=\\{\mathrm{IBM}\), U.S. Steel, Union Carbide, Boeing\\}. Which of these are proper subsets of \(A\) ?

According to data obtained from the National Weather Service, 376 of the 439 people killed by lightning in the United States between 1985 and 1992 were men. (Job and recreational habits of men make them more vulnerable to lightning.) Assuming that this trend holds in the future, what is the probability that a person killed by lightning a. Is a male? b. Is a female?

Evaluate the given expression. $$ P(n, n-2) $$

An exam consists of ten true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways can the exam be completed if a student can leave some questions unanswered because, say, a penalty is assessed for each incorrect answer?

In a poll conducted among 200 active investors, it was found that 120 use discount brokers, 126 use fullservice brokers, and 64 use both discount and full-service brokers. How many investors a. Use at least one kind of broker? b. Use exactly one kind of hroker? c. Use only discount brokers? d. Don't use a broker?

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets. $$ \\{1,2\\},\\{1,3,4\\},\\{4,6,8,10\\} $$

One light bulb is selected at random from a lot of 120 light bulbs, of which \(5 \%\) are defective. What is the probability that the light bulb selected is defective?

Evaluate the given expression. $$ C(n, n-2) $$

A warranty identification number for a certain product consists of a letter of the alphabet followed by a five-digit number. How many possible identification numbers are there if the first digit of the five-digit number must be nonzero?

CoMMUTER TRENDS Of 50 employees of a store located in downtown Boston, 18 people take the subway to work, 12 take the bus, and 7 take both the subway and the bus. How many employees a. Take the subway or the bus to work? b. Take only the bus to work? c. Take either the bus or the subway to work? d. Get to work by some other means?

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets. $$ \\{1,2,4\\},\\{a, b\\} $$

According to a survey of 176 retailers, \(46 \%\) of them use electronic tags as protection against shoplifting and employee theft. If one of these retailers is selected at random, what is the probability that he or she uses electronic tags as antitheft devices?

Let \(S=\\{a, b, c\\}\) be a sample space of an experiment with outcomes \(a, b\), and \(c\). List all the events of this experiment.

Classify each problem according to whether it involves a permutation or a combination. In how many ways can the letters of the word GLACIER be arranged?

In a state lottery, there are 15 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder can win more than one prize?

In a survey of 200 households regarding the ownership of desktop and laptop computers, the following information was obtained: 120 households own only desktop computers. 10 households own only laptop computers. 40 households own neither desktop nor laptop computers. How many households own both desktop and laptop computers?

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets. $$ \text { \\{Jill, John, Jack\\}, \\{Susan, Sharon\\} } $$

Explain why the statement is incorrect. There are eight grades in Garfield Elementary School. If a student is selected at random from the school, then the probability that the student is in the first grade is \(\frac{1}{8}\).

If a ball is selected at random from an urn containing three red balls, two white balls, and five blue balls, what is the probability that it will be a white ball?

Let \(S=\\{1,2,3\\}\) be a sample space associated with an experiment. a. List all events of this experiment. b. How many subsets of \(S\) contain the number 3 ? c. How many subsets of \(S\) contain either the number 2 or the number 3 ?

Classify each problem according to whether it involves a permutation or a combination. A four-member executive committee is to be formed from a twelve-member board of directors. In how many ways can it be formed?

a. How many seven-digit telephone numbers are possible if the first digit must be nonzero? b. How many direct-dialing numbers for calls within the United States and Canada are possible if each number consists of a 1 plus a three-digit area code (the first digit of which must be nonzero) and a number of the type described in part (a)?

In a survey of 400 households regarding the ownership of VCRs and DVD players, the following data were obtained: 360 households own one or more VCRs. 170 households own one or more VCRs and one or more DVD players. 19 households do not own a VCR or a DVD player. How many households own only one or more DVD players?