Chapter 8

An experiment consists of two independent trials. The outcomes of the first trial are \(A, B\), and \(C\), with probabilities of occurring equal to \(.2, .5\), and \(.3\), respectively. The outcomes of the second trial are \(E\) and \(F\), with probabilities of occurring equal to \(.6\) and \(.4 .\) Draw a tree diagram representing this experiment. Use this diagram to find a. \(P(B)\) b. \(P(F \mid B)\) c. \(P(B \cap F)\) d. \(P(F)\) e. Does \(P(B \cap F)=P(B) \cdot P(F) ?\) f. Are \(B\) and \(F\) independent events?

Find the probability that a family with three children will have the given composition. At least one girl

Rosa Walters is considering investing $$\$ 10,000$$ in two mutual funds. The anticipated returns from price appreciation and dividends (in hundreds of dollars) are described by the following probability distributions: $$\begin{array}{l}\text { Mutual Fund } \overline{\mathrm{A}}\\\\\begin{array}{rc} \hline \text { Returns } & \text { Probability } \\\\\hline-4 & .2 \\ \hline 8 & .5 \\\\\hline 10 & .3 \\\\\hline\end{array}\end{array}$$ $$\begin{array}{l}\text { Mutual Fund } \bar{B}\\\\\begin{array}{cc} \hline \text { Returns } & \text { Probability } \\\hline-2 & .2 \\ \hline 6 & .4 \\\\\hline 8 & .4 \\ \hline\end{array}\end{array}$$ a. Compute the mean and variance associated with the returns for each mutual fund. b. Which investment would provide Rosa with the higher expected return (the greater mean)? c. In which investment would the element of risk be less (that is, which probability distribution has the smaller variance)?

In a lottery, 5000 tickets are sold for $$\$ 1$$ each. One first prize of $$\$ 2000,1$$ second prize of $$\$ 500,3$$ third prizes of $$\$ 100$$, and 10 consolation prizes of \(\$ 25\) are to be awarded. What are the expected net earnings of a person who buys one ticket?

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. Represent the probabilities associated with this two-stage experiment in the form of a tree diagram.

A pair of fair dice is rolled. Let \(E\) denote the event that the number falling uppermost on the first die is 5 , and let \(F\) denote the event that the sum of the numbers falling uppermost is 10 . a. Compute \(P(F)\). b. Compute \(P(E \cap F)\). c. Compute \(P(F\lfloor E)\). d. Compute \(P(E)\). e. Are \(E\) and \(F\) independent events?

Find the probability that a family with three children will have the given composition. No girls

The distribution of the number of chocolate chips \((x)\) in a cookie is shown in the following table. Find the mean and the variance of the number of chocolate chips in a cookie. $$\begin{array}{llll}\hline x & 0 & 1 & 2 \\\\\hline P(X=x) & .01 & .03 & .05 \\\ \hline\end{array}$$ $$\begin{array}{llll}\hline \boldsymbol{x} & 3 & 4 & 5 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .11 & .13 & .24 \\ \hline\end{array}$$ $$\begin{array}{llll}\hline x & 6 & 7 & 8 \\\\\hline P(X=x) & .22 & .16 & .05 \\\ \hline\end{array}$$

A man wishes to purchase a 5 -yr term-life insurance policy that will pay the beneficiary $$\$ 20,000$$ in the event that the man's death occurs during the next 5 yr. Using life insurance tables, he determines that the probability that he will live another \(5 \mathrm{yr}\) is \(.96\). What is the minimum amount that he can expect to pay for his premium?

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was white given that the second ball drawn was white?

A pair of fair dice is rolled. Let \(E\) denote the event that the number falling uppermost on the first die is 4 , and let \(F\) denote the event that the sum of the numbers falling uppermost is 6 . a. Compute \(P(F)\). b. Compute \(P(E \cap F)\). c. Compute \(P(F \mid E)\). d. Compute \(P(E)\). e. Are \(E\) and \(F\) independent events?

Find the probability that a family with three children will have the given composition. The two oldest children are girls.

A woman purchased a $$\$ 10,000$$, 1-yr term-life insurance policy for $$\$ 130$$. Assuming that the probability that she will live another year is \(.992\), find the company's expected gain.

Two dice are rolled. Let the random variable \(X\) denote the number that falls uppermost on the first die, and let \(Y\) denote the number that falls uppermost on the second die. a. Find the probability distributions of \(X\) and \(Y\). b. Find the probability distribution of \(X+Y\).

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was black given that the second ball drawn was white?

A pair of fair dice is rolled. What is the probability that the sum of the numbers falling uppermost is less than 9, given that at least one of the numbers is a 6 ?

As a fringe benefit, Dennis Taylor receives a $$\$ 25,000$$ life insurance policy from his employer. The probability that Dennis will live another year is \(.9935\). If he purchases the same coverage for himself, what is the minimum amount that he can expect to pay for the policy?

A survey of 1000 families was conducted by the Public Housing Authority in a certain community to determine the distribution of families by size. The results follow: $$\begin{array}{lccccccc}\hline \text { Family Size } & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \text { Frequency of } & & & & & & & \\ text { Occurrence } & 350 & 200 & 245 & 125 & 66 & 10 & 4 \\ \hline\end{array}$$ a. Find the probability distribution of the random variable \(X\), where \(X\) denotes the number of persons in a randomly chosen family. b. Draw the histogram corresponding to the probability distribution found in part (a).

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was black given that the second ball drawn was black?

A pair of fair dice is rolled. What is the probability that the number landing uppermost on the first die is a 4 if it is known that the sum of the numbers landing uppermost is \(7 ?\)

Jacobs \& Johnson, an accounting firm, employs 14 accountants, of whom 8 are CPAs. If a delegation of 3 accountants is randomly selected from the firm to attend a conference, what is the probability that 3 CPAs will be selected?

The accompanying data were obtained in a study conducted by the manager of SavMore Supermarket. In this study, the number of customers waiting in line at the express checkout at the beginning of each 3 -min interval between 9 a.m. and 12 noon on Saturday was observed. $$\begin{array}{llllll}\hline \text { Customers } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Frequency of } & & & & & \\ \text { Occurrence } & 1 & 4 & 2 & 7 & 14 \\ \hline\end{array}$$ $$\begin{array}{lllllll}\hline \text { Customers } & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline \text { Frequency of } & & & & & & \\\\\text { Occurrence } & 8 & 10 & 6 & 3 & 4 & 1 \\\\\hline\end{array}$$ a. Find the probability distribution of the random variable \(X\), where \(X\) denotes the number of customers observed waiting in line. b. Draw the histogram representing the probability distribution.

A survey was conducted by the market research department of the National Real Estate Company among 500 prospective buyers in a large metropolitan area to determine the maximum price a prospective buyer would be willing to pay for a house. From the data collected, the distribution that follows was obtained. Compute the mean, variance, and standard deviation of the maximum price \(x\) (in thousands of dollars) that these buyers were willing to pay for a house. $$ \begin{array}{ll} \hline \text { Maximum Price } & \\ \text { Considered, } x & P(X=x) \\ \hline 280 & \frac{10}{500} \\ \hline 290 & \frac{20}{500} \\ \hline 300 & \frac{75}{500} \\ \hline 310 & \frac{85}{506} \\ \hline 320 & \frac{70}{500} \\ \hline 350 & \frac{90}{500} \\ \hline 380 & \frac{90}{500} \\ \hline 400 & \frac{55}{500} \\ \hline 450 & \frac{5}{500} \\ \hline\end{array}$$

Max built a spec house at a cost of $$\$ 450,000 .$$ He estimates that he can sell the house for $$\$ 580,000, \$$ 570,000\(, or \)\$ 560,000\(, with probabilities \).24\( \).40\(, and \).36$, respectively. What is Max's expected profit?

The 1992 U.S. Senate was composed of 57 Democrats and 43 Republicans. Of the Democrats, 38 served in the military, whereas 28 of the Republicans had seen military service. If a senator selected at random had served in the military, what is the probability that he or she was Republican?

A pair of fair dice is rolled. Let \(E\) denote the event that the number landing uppermost on the first die is a 3 , and let \(F\) denote the event that the sum of the numbers landing uppermost is 7. Determine whether \(E\) and \(F\) are independent events.

Two light bulbs are selected at random from a lot of 24 , of which 4 are defective. What is the probability that a. Both of the light bulbs are defective? b. At least 1 of the light bulbs is defective?

A study of the records of 85,000 apartment units in the greater Boston area revealed the following data: $$\begin{array}{llllll}\hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Average Rent, } \$ & 1352 & 1336 & 1317 & 1308 & 1355 \\\\\hline\end{array}$$ Find the average of the average rent for the \(5 \mathrm{yr}\) in question. What is the standard deviation for these data?

The proprietor of Midland Construction Company has to decide between two projects. He estimates that the first project will yield a profit of $$\$ 180,000$$ with a probability of \(.7\) or a profit of $$\$ 150,000$$ with a probability of \(.3\); the second project will yield a profit of $$\$ 220,000$$ with a probability of \(.6\) or a profit of $$\$ 80,000$$ with a probability of \(.4\). Which project should the proprietor choose if he wants to maximize his expected profit?

The interest rates paid by 30 financial institutions on a certain day for money market deposit accounts are shown in the accompanying table: $$\begin{array}{lcccc}\hline \text { Rate, \% } & 6 & 6.25 & 6.55 & 6.56 \\ \hline \text { Institutions } & 1 & 7 & 7 & 1 \\ \hline\end{array}$$ $$\begin{array}{lcccc} \hline \text { Rate, } \% & 6.58 & 6.60 & 6.65 & 6.85 \\\\\hline \text { Institutions } & 1 & 8 & 3 & 2 \\ \hline\end{array}$$ Let the random variable \(X\) denote the interest rate paid by a randomly chosen financial institution on its money market deposit accounts and find the probability distribution associated with these data.

In a survey of 2000 adults \(50 \mathrm{yr}\) and older of whom \(60 \%\) were retired and \(40 \%\) were preretired, the following question was asked: Do you expect your income needs to vary from year to year in retirement? Of those who were retired, \(33 \%\) answered no, and \(67 \%\) answered yes. Of those who were pre-retired, \(28 \%\) answered no, and \(72 \%\) answered yes. If a respondent in the survey was selected at random and had answered yes to the question, what is the probability that he or she was retired?

A customer at Cavallaro's Fruit Stand picks a sample of 3 oranges at random from a crate containing 60 oranges, of which 4 are rotten. What is the probability that the sample contains 1 or more rotten oranges?

A study of the records of 85,000 apartment units in the greater Boston area revealed the following data: $$\begin{array}{llllll}\hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Occupancy } & & & & & \\\\\text { Rate, \% } & 95.6 & 94.7 & 95.2 & 95.1 & 96.1 \\\\\hline\end{array}$$ Find the average occupancy rate for the 5 yr in question. What is the standard deviation for these data?

The management of MultiVision, a cable TV company, intends to submit a bid for the cable television rights in one of two cities, \(A\) or \(B\). If the company obtains the rights to city A, the probability of which is .2, the estimated profit over the next \(10 \mathrm{yr}\) is $$\$ 10$$ million: if the company obtains the rights to city \(\mathrm{B}\), the probability of which is \(.3\), the estimated profit over the next 10 yr is $$\$ 7$$ million. The cost of submitting a bid for rights in city \(\mathrm{A}\) is $$\$ 250,000$$ and that in city B is $$\$ 200,000$$. By comparing the expected profits for each venture, determine whether the company should bid for the rights in city A or city B.

After the private screening of a new television pilot, audience members were asked to rate the new show on a scale of 1 to 10 ( 10 being the highest rating). From a group of 140 people, the following responses were obtained: $$\begin{array}{lllllllllll}\hline \text { Rating } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \text { Frequency of } & & & & & & & & & & \\\\\text { Occurrence } & 1 & 4 & 3 & 11 & 23 & 21 & 28 & 29 & 16 & 4 \\\\\hline\end{array}$$ Let the random variable \(X\) denote the rating given to the show by a randomly chosen audience member. Find the probability distribution associated with these data.

An experiment consists of randomly selecting one of three coins, tossing it, and observing the outcome-heads or tails. The first coin is a two-headed coin, the second is a biased coin such that \(P(\mathrm{H})=.75\), and the third is a fair coin. a. What is the probability that the coin that is tossed will show heads? b. If the coin selected shows heads, what is the probability that this coin is the fair coin?

A card is drawn from a well-shuffled deck of 52 playing cards. Let \(E\) denote the event that the card drawn is black and let \(F\) denote the event that the card drawn is a spade. Determine whether \(E\) and \(F\) are independent events. Give an intuitive explanation for your answer.

A shelf in the Metro Department Store contains 80 colored ink cartridges for a popular ink-jet printer. Six of the cartridges are defective. If a customer selects 2 cartridges at random from the shelf, what is the probability that a. Both are defective? b. At least 1 is defective?

The following table gives the 2002 age distribution of the U.S. population: $$\begin{array}{lcccccc}\hline \text { Group } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \begin{array}{l}\text { Age } \\\\\text { (in years) } \end{array} & \text { Under } 5 & 5-19 & 20-24 & 25-44 & 45-64 & 65 \text { and over } \\\\\hline \begin{array}{l} \text { Number (in } \\\\\text { thousands) }\end{array} & 19,527 & 59,716 & 18,611 & 83,009 & 66,088 & 33,590 \\ \hline\end{array}$$ Let the random variable \(X\) denote a randomly chosen age group within the population. Find the probability distribution associated with these data.

The following table gives the scores of 30 students in a mathematics examination: $$\begin{array}{lccccc}\hline \text { Scores } & 90-99 & 80-89 & 70-79 & 60-69 & 50-59 \\ \hline \text { Students } & 4 & 8 & 12 & 4 & 2 \\ \hline\end{array}$$ Find the mean and the standard deviation of the distribution of the given data.

Roger Hunt intends to purchase one of two car dealerships currently for sale in a certain city. Records obtained from each of the two dealers reveal that their weekly volume of sales, with corresponding probabilities, are as follows: $$\begin{array}{l}\text { Dahl Motors }\\\ \begin{array}{lcccc}\hline \text { Cars Sold/Week } & 5 & 6 & 7 & 8 \\ \hline \text { Probability } & .05 & .09 & .14 & .24 \\\\\hline \end{array}\end{array}$$ $$\begin{array}{l}\text { Farthington Auto Sales }\\\ \begin{array}{lcccccc}\hline \text { Cars Sold/Week } & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline \text { Probability } & .08 & .21 & .31 & .24 & .10 & .06 \\\\\hline \end{array}\end{array}$$ The average profit/car at Dahl Motors is \(\$ 362\), and the average profit/car at Farthington Auto Sales is \(\$ 436\). a. Find the average number of cars sold each week at each dealership. b. If Roger's objective is to purchase the dealership that generates the higher weekly profit, which dealership should he purchase? (Compare the expected weekly profit for each dealership.)

A card is drawn from a well-shuffled deck of 52 playing cards. Let \(E\) denote the event that the card drawn is an ace and let \(F\) denote the event that the card drawn is a diamond. Determine whether \(E\) and \(F\) are independent events. Give an intuitive explanation for your answer.

Electronic baseball games manufactured by Tempco Electronics are shipped in lots of 24 . Before shipping, a quality-control inspector randomly selects a sample of 8 from each lot for testing. If the sample contains any defective games, the entire lot is rejected. What is the probability that a lot containing exactly 2 defective games will still be shipped?

The percentage of Boston homicide cases solved each year from 2000 through 2006 is summarized in the following table: $$\begin{array}{lccccccc} \hline \text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Percent } & 49 & 50 & 70 & 64 & 36 & 29 & 38 \\ \hline\end{array}$$ Find the average percent of Boston homicide cases solved per year for 2000 through \(2006 .\) What is the standard deviation for these data?

Sally Leonard, a real estate broker. is relocating in a large metropolitan area where she has received job offers from realty company A and realty company B. The number of houses she expects to sell in a year at each firm and the associated probabilities are shown in the following tables. The average price of a house in the locale of company \(\mathrm{A}\) is $$\$ 308,000$$, whereas the average price of a house in the locale of company \(\mathrm{B}\) is $$\$ 474,000$$. If Sally will receive a \(3 \%\) commission on sales at both companies, which job offer should she accept to maximize her expected yearly commission?

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. Suppose \(X\) is a finite discrete random variable assuming the values \(x_{1}, x_{2}, \ldots, x_{n}\) and associated probabilities \(p_{1}\), \(p_{2}, \ldots, p_{n} .\) Then \(p_{1}+p_{2}+\cdots+p_{n}=1\).

If a certain disease is present, then a blood test will reveal it \(95 \%\) of the time. But the test will also indicate the presence of the disease \(2 \%\) of the time when in fact the person tested is free of that disease; that is, the test gives a false positive \(2 \%\) of the time. If \(0.3 \%\) of the general population actually has the disease, what is the probability that a person chosen at random from the population has the disease given that he or she tested positive?

The probability that a battery will last \(10 \mathrm{hr}\) or more is \(.80\), and the probability that it will last \(15 \mathrm{hr}\) or more is .15. Given that a battery has lasted \(10 \mathrm{hr}\), find the probability that it will last \(15 \mathrm{hr}\) or more.

The number of married men (in thousands) between the ages of 20 and 44 in the United States in 1998 is given in the following table: $$\begin{array}{lccccc} \hline \text { Age } & 20-24 & 25-29 & 30-34 & 35-39 & 40-44 \\ \hline \text { Men } & 1332 & 4219 & 6345 & 7598 & 7633 \\\\\hline\end{array}$$ Find the mean and the standard deviation of the given data.

Bob, the proprietor of Midway Lumber, bases his projections for the annual revenues of the company on the performance of the housing market. He rates the performance of the market as very strong, strong, normal, weak, or very weak. For the next year, Bob estimates that the probabilities for these outcomes are $$18, .27$$, $$.42, .10$$, and $$.03$$, respectively. He also thinks that the revenues corresponding to these outcomes are $$\$ 20, \$$ 18.8$$, $$\$ 16.2, \$$ 14$$, and $$\$ 12$$ million, respectively. What is Bob's expected revenue for next year?