Chapter 8

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 area of a histogram associated with a probability distribution is a number between 0 and 1 .

In a survey to determine the opinions of Americans on health insurers, 400 baby boomers and 600 pre-boomers were asked this question: Do you believe that insurers are very responsible for high health costs? Of the baby boomers, 212 answered in the affirmative, whereas 198 of the pre-boomers answered in the affirmative. If a respondent chosen at random from those surveyed answered the question in the affirmative, what is the probability that he or she is a baby boomer? A pre-boomer?

Two cards are drawn without replacement from a wellshuffled deck of 52 playing cards. a. What is the probability that the first card drawn is a heart? b. What is the probability that the second card drawn is a heart if the first card drawn was not a heart? c. What is the probability that the second card drawn is a heart if the first card drawn was a heart?

The City Housing Authority has received 50 applications from qualified applicants for eight lowincome apartments. Three of the apartments are on the north side of town, and five are on the south side. If the apartments are to be assigned by means of a lottery, what is the probability that a. A specific qualified applicant will be selected for one of these apartments? b. Two specific qualified applicants will be selected for apartments on the same side of town?

The total number of pieces of mail delivered (in billions) each year from 2002 through 2006 is given in the following table: $$\begin{array}{lccccc}\hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Number } & 203 & 202 & 206 & 212 & 213 \\ \hline\end{array}$$ What is the average total number of pieces of mail delivered from 2002 through 2006 ? What is the standard deviation for these data?

Maria sees the growth of her business for the upcoming year as being tied to the gross domestic product (GDP). She believes that her business will grow (or contract) at the rate of \(5 \%, 4.5 \%, 3 \%, 0 \%\), or \(-0.5 \%\) per year if the GDP grows (or contracts) at the rate of between 2 and \(2.5 \%\), between \(1.5\) and \(2 \%\), between 1 and \(1.5 \%\), between 0 and \(1 \%\), and between \(-1\) and \(0 \%\), respectively. Maria has decided to assign a probability of \(.12, .24, .40, .20\), and \(.04\), respectively, to each outcome. At what rate does Maria expect her business to grow next year?

A halogen desk lamp produced by Luminar was found to be defective. The company has three factories where the lamps are manufactured. The percentage of the total number of halogen desk lamps produced by each factory and the probability that a lamp manufactured by that factory is defective are shown in the accompanying table. What is the probability that the defective lamp was manufactured in factory III? $$\begin{array}{ccc}\hline & & \text { Percent of } & \text { Probability of } \\\\\text { Factory } & \text { Total Production } & \begin{array}{c}\text { Defective } \\\\\text { Component }\end{array} \\ \hline \text { I } & 35 & .015 \\\\\hline \text { II } & 35 & .01 \\ \hline \text { III } & 30 & .02 \\\\\hline\end{array}$$

Five black balls and four white balls are placed in an urn. Two balls are then drawn in succession. What is the probability that the second ball drawn is a white ball if a. The second ball is drawn without replacing the first? b. The first ball is replaced before the second is drawn?

A student studying for a vocabulary test knows the meanings of 12 words from a list of 20 words. If the test contains 10 words from the study list, what is the probability that at least 8 of the words on the test are words that the student knows?

In a survey, consumers were asked how many television sets they have in their home. The results are summarized in the following table: $$\begin{array}{lccccc}\hline \text { TVs } & 1 & 2 & 3 & 4 & 5 \\\\\hline \text { Respondents, } \% & 13.9 & 26.5 & 28.6 & 14.8 & 16.2 \\\\\hline\end{array}$$\ Find the average number of TVs in the home of the respondents. What is the standard deviation for these data?

Suppose the probability that it will rain tomorrow is . 3 . a. What are the odds that it will rain tomorrow? b. What are the odds that it will not rain tomorrow?

A survey involving 400 likely Democratic voters and 300 likely Republican voters asked the question: Do you support or oppose legislation that would require registration of all handguns? The following results were obtained: $$\begin{array}{lcc}\hline \text { Answer } & \text { Democrats, \% } & \text { Republicans, \% } \\\\\hline \text { Support } & 77 & 59 \\\\\hline \text { Oppose } & 14 & 31 \\\\\hline \text { Don't know/refused } & 9 & 10 \\\\\hline\end{array}$$ If a randomly chosen respondent in the survey answered "oppose," what is the probability that he or she is a likely Democratic voter?

A tax specialist has estimated that the probability that a tax return selected at random will be audited is .02. Furthermore, he estimates that the probability that an audited return will result in additional assessments being levied on the taxpayer is .60. What is the probability that a tax return selected at random will result in additional assessments being levied on the taxpayer?

Four different written driving tests are administered by the Motor Vehicle Department. One of these four tests is selected at random for each applicant for a driver's license. If a group consisting of two women and three men apply for a license, what is the probability that a. Exactly two of the five will take the same test? b. The two women will take the same test?

The number of average hours worked per year per worker in the United States and five European countries in 2002 is given in the following table: $$\begin{array}{lcccccc}\hline & & & \text { Great } & & \text { West } & \\\\\text { Country } & \text { U.S. } & \text { Spain } & \text { Britain } & \text { France } & \text { Germany } & \text {Norway } \\ \hline \text { Average } & & & & & & \\\\\text { Hours } & 1815 & 1807 & 1707 & 1545 & 1428 & 1342 \\ \text { Werked } & & & & & & \\ \hline\end{array}$$ Find the average of the average hours worked per worker in 2002 for workers in the six countries. What is the standard deviation for these data?

In American roulette, as described in Example 6, a player may bet on a split (two adjacent numbers). In this case, if the player bets $$\$ 1$$ and either number comes up, the player wins $$\$ 17$$ and gets his $$\$ 1$$ back. If neither comes up, he loses his $$\$ 1$$ bet. Find the expected value of the winnings on a $$\$ 1$$ bet placed on a split.

A study conducted by the Metro Housing Agency in a midwestern city revealed the following information concerning the age distribution of renters within the city. $$\begin{array}{lcc}\hline & \text { Group } \\\\\text { Age } & \text { Adult Population, \% } & \text { Who Are Renters, \% } \\\\\hline 21-44 & 51 & 58 \\\\\hline 45-64 & 31 & 45 \\\\\hline 65 \text { and over } & 18 & 60 \\ \hline\end{array}$$ a. What is the probability that an adult selected at random from this population is a renter? b. If a renter is selected at random, what is the probability that he or she is in the \(21-44\) age bracket? c. If a renter is selected at random, what is the probability that he or she is 45 yr of age or older?

At a certain medical school, \(\frac{1}{7}\) of the students are from a minority group. Of those students who belong to a minority group, \(\frac{1}{3}\) are black. a. What is the probability that a student selected at random from this medical school is black? b. What is the probability that a student selected at random from this medical school is black if it is known that the student is a member of a minority group?

The number of Americans without health insurance, in millions, from 1995 through 2002 is summarized in the following table: $$\begin{array}{lllllllll}\hline \text { Year } & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 & 2002 \\ \hline \text { Number } & 40.7 & 41.8 & 43.5 & 44.5 & 40.2 & 39.9 & 41.2 & 43.6 \\\\\hline\end{array}$$ Find the average number of Americans without health insurance in the period from 1995 through 2002 . What is the standard deviation for these data?

If a player placed a $$\$ 1$$ bet on \(r e d\) and a $$\$ 1$$ bet on black in a single play in American roulette, what would be the expected value of his winnings?

The estimated probability that a brand-A, a brand-B, and a brand-C plasma TV will last at least \(30,000 \mathrm{hr}\) is \(.90, .85\), and \(.80\), respectively. Of the 4500 plasma TVs that Ace TV sold in a certain year, 1000 were brand A, 1500 were brand \(\mathrm{B}\), and 2000 were brand \(\mathrm{C}\). If a plasma TV set sold by Ace TV that year is selected at random and is still working after \(30,000 \mathrm{hr}\) of use a. What is the probability that it was a brand-A TV? b. What is the probability that it was not a brand-A TV?

In a survey of 1000 eligible voters selected at random, it was found that 80 had a college degree. Additionally, it was found that \(80 \%\) of those who had a college degree voted in the last presidential election. whereas \(55 \%\) of the people who did not have a college degree voted in the last presidential election. Assuming that the poll is representative of all eligible voters, find the probability that an eligible voter selected at random a. Had a college degree and voted in the last presidential election. b. Did not have a college degree and did not vote in the last presidential election. c. Voted in the last presidential election. d. Did not vote in the last presidential election.

In the game of blackjack, a 2-card hand consisting of an ace and a face card or a 10 is called a blackjack. a. If a player is dealt 2 cards from a standard deck of 52 well-shuffled cards, what is the probability that the player will receive a blackjack? b. If a player is dealt 2 cards from 2 well-shuffled standard decks, what is the probability that the player will receive a blackiack?

In European roulette, the wheel is divided into 37 compartments numbered 1 through 36 and 0 . (In American roulette there are 38 compartments numbered 1 through 36,0 , and 00 .) Find the expected value of the winnings on a \(\$ 1\) bet placed on \(r e d\) in European roulette.

A survey involving 400 likely Democratic voters and 300 likely Republican voters asked the question: Do you support or oppose legislation that would require trigger locks on guns, to prevent misuse by children? The following results were obtained: $$\begin{array}{lcc}\hline \text { Answer } & \text { Democrats, \% } & \text { Republicans, \% } \\\\\hline \text { Support } & 88 & 71 \\\\\hline \text { Oppose } & 7 & 20 \\\\\hline \text { Don't know/refused } & 5 & 9 \\ \hline\end{array}$$ If a randomly chosen respondent in the survey answered "support," what is the probability that he or she is a likely Republican voter?

Three cards are drawn without replacement from a wellshuffled deck of 52 playing cards. What is the probability that the third card drawn is a diamond?

The probability of an event \(E\) occurring is \(.8\). What are the odds in favor of \(E\) occurring? What are the odds against \(E\) occurring?

A coin is tossed three times. What is the probability that the coin will land heads a. At least twice? b. On the second toss, given that heads were thrown on the first toss? C. On the third toss, given that tails were thrown on the first toss?

In 1959 a world record was set for the longest run on an ungaffed (fair) roulette wheel at the El San Juan Hotel in Puerto Rico. The number 10 appeared six times in a row. What is the probability of the occurrence of this event? (Assume that there are 38 equally likely outcomes consisting of the numbers \(1-36,0\), and 00 .)

The seasonally adjusted annualized sales rate for U.S. cars and light trucks, in millions of units, for May 2003 through April 2004 are given in the following tables: $$\begin{array}{l}2003\\\\\begin{array}{cccccccc}\hline \mathrm{M} & \mathrm{J} & \mathrm{J} & \mathrm{A} & \mathrm{S} & \mathrm{O} & \mathrm{N} &\mathrm{D} \\ \hline 16.5 & 16.5 & 17.0 & 18.5 & 17.0 & 16.0 & 17.0 & 18.0 \\\\\hline \end{array}\end{array}$$ $$\begin{array}{l}2004\\\\\begin{array}{cccc}\hline \mathrm{J} & \mathrm{F} & \mathrm{M} & \mathrm{A} \\ \hline 16.3 & 16.5 & 16.8 & 16.5 \\\\\hline \end{array}\end{array}$$ What is the average seasonally adjusted annualized sales rate for U.S. motor vehicles for the period in question? What is the standard deviation for these data?

The probability of an event \(E\) not occurring is .6. What are the odds in favor of \(E\) occurring? What are the odds against \(E\) occurring?

Data compiled by the Department of Justice on the number of people arrested in a certain year for serious crimes (murder, forcible rape, robbery, and so on) revealed that \(89 \%\) were male and \(11 \%\) were female. Of the males, \(30 \%\) were under 18 , whereas \(27 \%\) of the females arrested were under 18 . a. What is the probability that a person arrested for a serious crime in that year was under 18 ? b. If a person arrested for a serious crime in that year is known to be under 18 , what is the probability that the person is female?

In a three-child family, what is the probability that all three children are girls given that at least one of the children is a girl? (Assume that the probability of a boy being born is the same as the probability of a girl being born.)

The percent of the voting age population who cast ballots in presidential elections from 1932 through 2000 are given in the following table: $$\begin{array}{llllllllll}\hline \begin{array}{l}\text { Election } \\ \text { Year }\end{array} & 1932 & 1936 & 1940 & 1944 & 1948 & 1952 & 1956 & 1960 & 1964 \\ \hline \text { Turnout, \% } & 53 & 57 & 59 & 56 & 51 & 62 & 59 & 59 & 62 \\ \hline\end{array}$$ $$\begin{array}{llllllllll}\hline \begin{array}{l}\text { Election } \\ \text { Year }\end{array} & 1968 & 1972 & 1976 & 1980 & 1984 & 1988 & 1992 & 1996 & 2000 \\\\\hline \text { Turnout } \% & 61 & 55 & 54 & 53 & 53 & 50 & 55 & 49 & 51 \\ \hline\end{array}$$ Find the mean and the standard deviation of the given data.

The odds in favor of an event \(E\) occurring are 9 to 7 . What is the probability of \(E\) occurring?

A medical test has been designed to detect the presence of a certain disease. Among those who have the disease, the probability that the disease will be detected by the test is .95. However, the probability that the test will erroneously indicate the presence of the disease in those who do not actually have it is .04. It is estimated that \(4 \%\) of the population who take this test have the disease. a. If the test administered to an individual is positive, what is the probability that the person actually has the disease? b. If an individual takes the test twice and both times the test is positive, what is the probability that the person actually has the disease? (Assume that the tests are independent.)

An automobile manufacturer obtains the microprocessors used to regulate fuel consumption in its automobiles from three microelectronic firms: A, B, and C. The quality-control department of the company has determined that \(1 \%\) of the microprocessors produced by firm A are defective, \(2 \%\) of those produced by firm \(B\) are defective, and \(1.5 \%\) of those produced by firm \(\mathrm{C}\) are defective. Firms \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) supply \(45 \%, 25 \%\), and \(30 \%\), respectively, of the microprocessors used by the company. What is the probability that a randomly selected automobile manufactured by the company will have a defective microprocessor?

A probability distribution has a mean of 42 and a standard deviation of \(2 .\) Use Chebychev's inequality to find a bound on the probability that an outcome of the experiment lies between a. 38 and 46 . b. 32 and 52 .

The odds against an event \(E\) occurring are 2 to \(3 .\) What is the probability of \(E\) not occurring?

Figures obtained from a city's police department seem to indicate that, of all motor vehicles reported as stolen, \(64 \%\) were stolen by professionals whereas \(36 \%\) were stolen by amateurs (primarily for joy rides). Of those vehicles presumed stolen by professionals, \(24 \%\) were recovered within \(48 \mathrm{hr}, 16 \%\) were recovered after \(48 \mathrm{hr}\), and \(60 \%\) were never recovered. Of those vehicles presumed stolen by amateurs, \(38 \%\) were recovered within \(48 \mathrm{hr}, 58 \%\) were recovered after \(48 \mathrm{hr}\), and \(4 \%\) were never recovered. a. Draw a tree diagram representing these data. b. What is the probability that a vehicle stolen by a professional in this city will be recovered within \(48 \mathrm{hr}\) ? c. What is the probability that a vehicle stolen in this city will never be recovered?

A probability distribution has a mean of 20 and a standard deviation of \(3 .\) Use Chebychev's inequality to find a bound on the probability that an outcome of the experiment lies between a. 15 and 25 . b. 10 and 30 .

Carmen, a computer sales representative, feels that the odds are 8 to 5 that she will clinch the sale of a minicomputer to a certain company. What is the (subjective) probability that Carmen will make the sale?

A study of the faculty at U.S. medical schools in 2006 revealed that \(32 \%\) of the faculty were women and \(68 \%\) were men. Of the female faculty, \(31 \%\) were full/associate professors, \(47 \%\) were assistant professors, and \(22 \%\) were instructors. Of the male faculty, \(51 \%\) were full/associate professors, \(37 \%\) were assistant professors, and \(12 \%\) were instructors. If a faculty member at a U.S. medical school selected at random holds the rank of full/associate professor, what is the probability that she is female?

The chief loan officer of La Crosse Home Mortgage Company summarized the housing loans extended by the company in 2007 according to type and term of the loan. Her list shows that \(70 \%\) of the loans were fixed-rate mortgages \((F), 25 \%\) were adjustable-rate mortgages \((A)\), and \(5 \%\) belong to some other category \((O)\) (mostly second trust-deed loans and loans extended under the graduated payment plan). Of the fixed-rate mortgages, \(80 \%\) were 30 -yr loans and \(20 \%\) were 15 -yr loans; of the adjustable-rate mortgages, \(40 \%\) were 30 -yr loans and \(60 \%\) were 15 -yr loans; finally, of the other loans extended, \(30 \%\) were 20 -yr loans, \(60 \%\) were 10 -yr loans, and \(10 \%\) were for a term of 5 yr or less. a. Draw a tree diagram representing these data. b. What is the probability that a home loan extended by La Crosse has an adjustable rate and is for a term of 15 yr? c. What is the probability that a home loan extended by La Crosse is for a term of 15 yr?

A probability distribution has a mean of 50 and a standard deviation of \(1.4\). Use Chebychev's inequality to find the value of \(c\) that guarantees the probability is at least \(96 \%\) that an outcome of the experiment lies between \(50-c\) and \(50+c .\)

Steffi feels that the odds in favor of her winning her tennis match tomorrow are 7 to \(5 .\) What is the (subjective) probability that she will win her match tomorrow?

In a study of the scientific research on soft drinks, juices, and milk, 50 studies were fully sponsored by the food industry, and 30 studies were conducted with no corporate ties. Of those that were fully sponsored by the food industry, \(14 \%\) of the participants found the products unfavorable, \(23 \%\) were neutral, and \(63 \%\) found the products favorable. Of those that had no industry funding, \(38 \%\) found the products unfavorable, \(15 \%\) were neutral, and \(47 \%\) found the products favorable. a. What is the probability that a participant selected at random found the products favorable? b. If a participant selected at random found the product favorable, what is the probability that he or she belongs to a group that participated in a corporate-sponsored study?

The admissions office of a private university released the following admission data for the preceding academic year: From a pool of 3900 male applicants, \(40 \%\) were accepted by the university and \(40 \%\) of these subsequently enrolled. Additionally, from a pool of 3600 female applicants, \(45 \%\) were accepted by the university and \(40 \%\) of these subsequently enrolled. What is the probability that a. A male applicant will be accepted by and subsequently will enroll in the university? b. A student who applies for admissions will be accepted by the university? c. A student who applies for admission will be accepted by the university and subsequently will enroll?

Suppose \(X\) is a random variable with mean \(\mu\) and standard deviation \(\sigma\). If a large number of trials is observed, at least what percentage of these values is expected to lie between \(\mu-2 \sigma\) and \(\mu+2 \sigma ?\)

If a sports forecaster states that the odds of a certain boxer winning a match are 4 to 3 , what is the (subjective) probability that the boxer will win the match?