Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 3

Elements of Chance: Probability Methods - all with Video Answers

Educators

Chapter Questions

Problem 1

For Exercises 3.1-3.4 use the sample space $S$ defined as follows:
$$
S=\left[E_1, E_2, E_3, E_4, E_5, E_6, E_7, E_8, E_9, E_{10}\right]
$$
Given $A=\left[E_1, E_3, E_6, E_9\right]$, define $\bar{A}$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 2

For Exercises 3.1-3.4 use the sample space $S$ defined as follows:
$$
S=\left[E_1, E_2, E_3, E_4, E_5, E_6, E_7, E_8, E_9, E_{10}\right]
$$
Given $A=\left[E_1, E_3, E_7, E_9\right]$ and $B=\left[E_2, E_3, E_8, E_9\right]$.
a. What is $A$ intersection $B$ ?
b. What is the union of $A$ and $B$ ?
c. Is the union of $A$ and $B$ collectively exhaustive?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 3

For Exercises 3.1-3.4 use the sample space $S$ defined as follows:
$$
S=\left[E_1, E_2, E_3, E_4, E_5, E_6, E_7, E_8, E_9, E_{10}\right]
$$
Given $\bar{A}=\left[E_1, E_3, E_7, E_9\right]$ and $\overline{B}=\left[E_2, E_3, E_8, E_9\right]$.
a. What is the intersection of $A$ intersection $B$ ?
b. What is the union of $A$ and $B$ ?
c. Is the union of $A$ and $B$ collectively exhaustive?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 4

For Exercises 3.1-3.4 use the sample space $S$ defined as follows:
$$
S=\left[E_1, E_2, E_3, E_4, E_5, E_6, E_7, E_8, E_9, E_{10}\right]
$$
Given $A=\left[E_3, E_5, E_{6,} E_{10}\right]$ and $B=\left[E_3, E_4, E_{60} E_9\right]$
a. What is the intersection of $A$ and $B$ ?
b. What is the union of $A$ and $B$ ?
c. Is the union of $A$ and $B$ collectively exhaustive?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 5

A corporation takes delivery of some new machinery that must be installed and checked before it becomes available to use. The corporation is sure that it will take no more than 7 days for this installation and check to take place. Let $A$ be the event "it will be more than 4 days before the machinery becomes available" and $B$ be the event "it will be less than 6 days before the machinery becomes available."
a. Describe the event that is the complement of event $A$.
b. Describe the event that is the intersection of events $A$ and $B$.
c. Describe the event that is the union of events $A$ and $B$.
d. Are events $A$ and $B$ mutually exclusive?
e. Are events $A$ and $B$ collectively exhaustive?
f. Show that $(A \cap B) \cup(\bar{A} \cap B)=B$.
g. Show that $A \cup(\bar{A} \cap B)=A \cup B$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 6

Consider Example 3.4, with the following four basic outcomes for the Dow Jones Industrial Average over two consecutive days:
$O_1$ : The Dow Jones average rises on both days.
$\mathrm{O}_2$ : The Dow Jones average rises on the first day but does not rise on the second day.
$\mathrm{O}_3$ : The Dow Jones average does not rise on the first day but rises on the second day.
$\mathrm{O}_4$ : The Dow Jones average does not rise on either day.
Let events $A$ and $B$ be the following:
$A$ : The Dow Jones average rises on the first day.
$B$ : The Dow Jones average rises on the second day.
a. Show that $(A \cap B) \cup(\bar{A} \cap B)=B$.
b. Show that $A \cup(\bar{A} \cap B)=A \cup B$.

Nick Johnson

Numerade Educator

Problem 7

Florin Frenti operates a small, used car lot that has three Mercedes $\left(M_1, M_2, M_3\right)$ and two Toyotas $\left(T_1, T_2\right)$. Two customers, Cezara and Anda, come to his lot, and each selects a car. The customers do not know each other, and there is no communication between them. Let the events $A$ and $B$ be defined as follows:
$A$ : The customers select at least one Toyota.
$B$ : The customers select two cars of the same model.
a. Identify all pairs of cars in the sample space.
b. Define event $A$.
c. Define event $B$.
d. Define the complement of $A$.
e. Show that $(A \cap B) \cup(\bar{A} \cap B)=B$.
f. Show that $A \cup(\bar{A} \cap B)=A \cup B$.

Doruk Isik

Numerade Educator

Problem 8

The sample space contains $5 \mathrm{As}$ and $7 \mathrm{Bs}$. What is the probability that a randomly selected set of 2 will include $1 \mathrm{~A}$ and $1 \mathrm{~B}$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 9

The sample space contains $6 \mathrm{As}$ and $4 \mathrm{Bs}$. What is the probability that a randomly selected set of 3 will include $1 \mathrm{~A}$ and $2 \mathrm{Bs}$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 10

The sample space contains 10 As and $6 \mathrm{Bs}$. What is the probability that a randomly selected set of 4 will include 2 As and $2 \mathrm{Bs}$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 11

In a city of 120,000 people there are 20,000 Norwegians. What is the probability that a randomly selected person from the city will be Norwegian?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 12

In a city of 180,000 people there are 20,000 legal immigrants from Latin America. What is the probability that a random sample of two people from the city will contain two legal immigrants from Latin America?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 13

A corporation has just received new machinery that must be installed and checked before it becomes operational. The accompanying table shows a manager's probability assessment for the number of days required before the machinery becomes operational.
$$
\begin{array}{lrrrrr}
\hline \text { Number of days } & 3 & 4 & 5 & 6 & 7 \\
\hline \text { Probability } & 0.08 & 0.24 & 0.41 & 0.20 & 0.07 \\
\hline
\end{array}
$$
Let $A$ be the event "it will be more than four days before the machinery becomes operational," and let $B$ be the event "it will be less than six days before the machinery becomes available."
a. Find the probability of event $A$.
b. Find the probability of event $B$.
c. Find the probability of the complement of event $A$.
d. Find the probability of the intersection of events $A$ and $B$.
e. Find the probability of the union of events $A$ and $B$.

Sheryl Ezze

Numerade Educator

Problem 14

On a sample of 1,500 people in Sydney, Australia, 89 have no credit cards (event $A$ ), 750 have one (event B), 450 have two (event $C$ ) and the rest have more than two (event $D$ ). On the basis of the data, calculate each of the following.
a. The probability of event $A$
b. The probability of event $D$
c. The complement of event $B$
d. The complement of event $C$
e. The probability of event $A$ or $D$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 15

A manager has available a pool of 8 employees who could be assigned to a project-monitoring task. 4 of the employees are women and 4 are men. 2 of the men are brothers. The manager is to make the assignment at random so that each of the 8 employees is equally likely to be chosen. Let $A$ be the event "chosen employee is a man" and $B$ the event "chosen employee is one of the brothers."
a. Find the probability of $A$.
b. Find the probability of $B$.
c. Find the probability of the intersection of $A$ and $B$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 16

If two events are mutually exclusive, we know that the probability of their union is the sum of their individual probabilities. However, this is not the case for events that are not mutually exclusive. Verify this assertion by considering the events $A$ and $B$ of Exercise 3.2.

Sheryl Ezze

Numerade Educator

Problem 17

A department store manager has monitored the number of complaints received per week about poor service. The probabilities for numbers of complaints in a week, established by this review, are shown in the following table. Let $A$ be the event "there will be at least one complaint in a week" and $B$ the event "there will be fewer than ten complaints in a week."
$$
\begin{array}{lcccccc}
\hline \begin{array}{l}
\text { Number of } \\
\text { complaints }
\end{array} & 0 & 1 \text { to } 3 & 4 \text { to } 6 & 7 \text { to } 9 & 10 \text { to } 12 & \begin{array}{c}
\text { More } \\
\text { than } 12
\end{array} \\
\hline \text { Probability } & 0.14 & 0.39 & 0.23 & 0.15 & 0.06 & 0.03 \\
\hline
\end{array}
$$}
a. Find the probability of $A$.
b. Find the probability of $B$.
c. Find the probability of the complement of $A$.
d. Find the probability of the union of $A$ and $B$.
e. Find the probability of the intersection of $A$ and $B$.
f. Are $A$ and $B$ mutually exclusive?
g. Are $A$ and $B$ collectively exhaustive?

Hoan Nguyen

Numerade Educator

Problem 18

A corporation receives a particular part in shipments of 100 . Research indicated the probabilities shown in the accompanying table for numbers of defective parts in a shipment.
$$
\begin{array}{lccccc}
\hline \text { Number } & 0 & 1 & 2 & 3 & >3 \text { defective } \\
\hline \text { Probability } & 0.29 & 0.36 & 0.22 & 0.10 & 0.03 \\
\hline
\end{array}
$$
a. What is the probability that there will be fewer than three defective parts in a shipment?
b. What is the probability that there will be more than one defective part in a shipment?
c. The five probabilities in the table sum to 1. Why must this be so?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 19

The probability of $A$ is 0.60 , the probability of $B$ is 0.45 , and the probability of either is 0.80 . What is the probability of both $A$ and $B$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 20

The probability of $A$ is 0.40 , the probability of $B$ is 0.45 , and the probability of either is 0.85 . What is the probability of both $A$ and $B$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 21

The probability of $A$ is 0.60 , the probability of $B$ is 0.40 , and the probability of either is 0.76 . What is the probability of both $A$ and $B$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 22

The probability of $A$ is 0.60 , the probability of $B$ is 0.45 , and the probability of both is 0.30 . What is the probability of either $A$ and $B$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 23

The probability of $A$ is 0.60 , the probability of $B$ is 0.45 , and the probability of both is 0.30 . What is the conditional probability of $A$, given $B$ ? Are $A$ and $B$ independent in a probability sense?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 24

The probability of $A$ is 0.80 , the probability of $B$ is 0.10 , and the probability of both is 0.08 . What is the conditional probability of $A$, given $B$ ? Are $A$ and $B$ independent in a probability sense?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 25

The probability of $A$ is 0.30 , the probability of $B$ is 0.40 and the probability of both is 0.30 . What is the conditional probability of $A$ given $B$ ? Are $A$ and $B$ independent in a probability sense?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 26

The probability of $A$ is 0.70 , the probability of $B$ is 0.80 , and the probability of both is 0.50 . What is the conditional probability of $A$, given $B$ ? Are $A$ and $B$ independent in a probability sense?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 27

A company knows that a rival is about to bring out a competing product. It believes that this rival has three possible packaging plans (superior, normal, and cheap) in mind and that all are equally likely. Also, there are three equally likely possible marketing strategies (intense media advertising, price discounts, and the use of a coupon to reduce the price of future purchases). What is the probability that the rival will employ superior packaging in conjunction with an intense media advertising campaign? Assume that packaging plans and marketing strategies are determined independently.

Sheryl Ezze

Numerade Educator

Problem 28

A financial analyst was asked to evaluate earnings prospects for seven corporations over the next year and to rank them in order of predicted earnings growth rates.
a. How many different rankings are possible?
b. If, in fact, a specific ordering is the result of a guess, what is the probability that this guess will turn out to be correct?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 29

A company has 50 sales representatives. It decides that the most successful representative during the previous year will be awarded a January vacation in Hawaii, while the second most successful will win a vacation in Las Vegas. The other representatives will be required to attend a conference on modern sales methods in Buffalo. How many outcomes are possible?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 30

A securities analyst claims that, given a specific list of 6 common stocks, it is possible to predict, in the correct order, the 3 that will perform best during the coming year. What is the probability of making the correct selection by chance?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 31

A student committee has 6 members-4 undergraduate and 2 graduate students. A subcommittee of 3 members is to be chosen randomly so that each possible combination of 3 of the 6 students is equally likely to be selected. What is the probability that there will be no graduate students on the subcommittee?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 32

The soccer league in 1 community has 5 teams. You are required to predict, in order, the top 3 teams at the end of the season. Ignoring the possibility of ties, calculate the number of different predictions you could make. What is the probability of making the correct prediction by chance?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 33

The senior management of a corporation has decided that in the future it wishes to divide its consulting budget between 2 firms. 8 firms are currently being considered for this work. How many different choices of 2 firms are possible?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 34

You are 1 of 7 female candidates auditioning for 2 parts-the heroine and her best friend-in a play. Before the auditions you know nothing of the other candidates, and you assume all candidates have equal chances for the parts.
a. How many distinct choices are possible for casting the two parts?
b. In how many of the possibilities in part (a) would you be chosen to play the heroine?
c. In how many of the possibilities in part (a) would you be chosen to play the best friend?
d. Use the results in parts (a) and (b) to find the probability that you will be chosen to play the heroine. Indicate a more direct way of finding this probability.
e. Use the results in parts (a), (b), and (c) to find the probability that you will be chosen to play 1 of the 2 parts. Indicate a more direct way of finding this probability.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 35

A work crew for a building project is to be made up of 2 craftsmen and 4 laborers selected from a total of 5 craftsmen and 6 laborers.
a. How many different combinations are possible?
b. The brother of one of the craftsmen is a laborer. If the crew is selected at random, what is the probability that both brothers will be selected?
c. What is the probability that neither brother will be selected?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 36

A mutual fund company has 6 funds that invest in the U.S. market and 4 that invest in international markets. A customer wants to invest in two U.S. funds and 2 international funds.
a. How many different sets of funds from this company could the investor choose?
b. Unknown to this investor, one of the U.S. funds and one of the international funds will seriously underperform next year. If the investor selects funds for purchase at random, what is the probability that at least one of the chosen funds will seriously underperform next year?

Sheryl Ezze

Numerade Educator

Problem 37

It was estimated that $30 \%$ of all seniors on a campus were seriously concerned about employment prospects, $25 \%$ were seriously concerned about grades, and $20 \%$ were seriously concerned about both. What is the probability that a randomly chosen senior from this campus is seriously concerned about at least one of these two things?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 38

A video movie store owner finds that $30 \%$ of the customers entering the store ask an assistant for help and that $20 \%$ of the customers make a purchase before leaving. It is also found that $15 \%$ of all customers both ask for assistance and make a purchase. What is the probability that a customer does at least one of these two things?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 39

A local public-action group solicits donations by telephone. For a particular list of prospects it was estimated that for any individual the probability was 0.05 of an immediate donation by credit card, 0.25 of no immediate donation but a request for further information through the mail, and 0.7 of no expression of interest. Information is mailed to all people requesting it, and it is estimated that $20 \%$ of these people will eventually donate. An operator makes a sequence of calls, the outcomes of which can be assumed to be independent.
a. What is the probability that no immediate creditcard donation will be received until at least four unsuccessful calls have been made?
b. What is the probability that the first call leading to any donation (either immediately or eventually after a mailing) is preceded by at least four unsuccessful calls?

Nick Johnson

Numerade Educator

Problem 40

A mail-order firm considers three possible events in filling an order:
$A$ : The wrong item is sent.
$B$ : The item is lost in transit.
$C$ : The item is damaged in transit.
Assume that $A$ is independent of both $B$ and $C$ and that $B$ and $C$ are mutually exclusive. The individual event probabilities are $P(A)=0.02, P(B)=0.01$, and $P(C)=0.04$. Find the probability that at least one of these foul-ups occurs for a randomly chosen order.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 41

A coach recruits for a college team a star player who is currently a high school senior. In order to play next year, the senior must both complete high school with adequate grades and pass a standardized test. The coach estimates that the probability the athlete will fail to obtain adequate high school grades is 0.02 , that the probability the athlete will not pass the standardized test is 0.15 , and that these are independent events. According to these estimates, what is the probability that this recruit will be eligible to play in college next year?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 42

Market research in a particular city indicated that during a week, $18 \%$ of all adults watch a television program oriented to business and financial issues, $12 \%$ read a publication oriented to these issues, and $10 \%$ do both.
a. What is the probability that an adult in this city who watches a television program oriented to business and financial issues reads a publication oriented to these issues?
b. What is the probability that an adult in this city who reads a publication oriented to business and financial issues watches a television program oriented to these issues?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 43

In Sipadan, Malaysia, there is a national park where up to 100 dolphins can be found. Suppose we randomly select two of them in one draw.
a. What is the probability that we pick two females, knowing that there are only 10 females in all?
b. What is the probability of getting two males instead?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 44

An analyst is presented with lists of 4 stocks and 5 bonds. He is asked to predict, in order, the 2 stocks that will yield the highest return over the next year and the 2 bonds that will have the highest return over the next year. Suppose that these predictions are made randomly and independently of each other. What is the probability that the analyst will be successful in at least 1 of the 2 tasks?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 45

A bank classifies borrowers as high risk or low risk. Only $15 \%$ of its loans are made to those in the highrisk category. Of all its loans, $5 \%$ are in default, and $40 \%$ of those in default were made to high-risk borrowers. What is the probability that a high-risk borrower will default?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 46

A conference began at noon with two parallel sessions. The session on portfolio management was attended by $40 \%$ of the delegates, while the session on chartism was attended by $50 \%$. The evening session consisted of a talk titled "Is the Random Walk Dead?" This was attended by $80 \%$ of all delegates.
a. If attendance at the portfolio management session and attendance at the chartism session are mutually exclusive, what is the probability that a randomly chosen delegate attended at least one of these sessions?
b. If attendance at the portfolio management session and attendance at the evening session are statistically independent, what is the probability that a randomly chosen delegate attended at least one of these sessions?
c. Of those attending the chartism session, $75 \%$ also attended the evening session. What is the probability that a randomly chosen delegate attended at least one of these two sessions?

Sheryl Ezze

Numerade Educator

Problem 47

A stock market analyst claims expertise in picking stocks that will outperform the corresponding industry norms. This analyst is presented with a list of 5 high-technology stocks and a list of 5 airline stocks, and she is invited to nominate, in order, the 3 stocks that will do best on each of these 2 lists over the next year. The analyst claims that success in just 1 of these 2 tasks would be a substantial accomplishment. If, in fact, the choices are made randomly and independently, what is the probability of success in at least 1 of the 2 tasks merely by chance? Given this result, what do you think of the analyst's claim?

Nick Johnson

Numerade Educator

Problem 48

A quality-control manager found that $30 \%$ of workrelated problems occurred on Mondays and that $20 \%$ occurred in the last hour of a day's shift. It was also found that $4 \%$ of worker-related problems occurred in the last hour of Monday's shift.
a. What is the probability that a worker-related problem that occurs on a Monday does not occur in the last hour of the day's shift?
b. Are the events "problem occurs on Monday" and "problem occurs in the last hour of the day's shift" statistically independent?

Sheryl Ezze

Numerade Educator

Problem 49

A corporation was concerned with the basic educational skills of its workers and decided to offer a selected group of them separate classes in reading and practical mathematics. Of these workers, $40 \%$ signed up for the reading classes and $50 \%$ for the practical mathematics classes. Of those signing up for the reading classes $30 \%$ signed up for the mathematics classes.
a. What is the probability that a randomly selected worker signed up for both classes?
b. What is the probability that a randomly selected worker who signed up for the mathematics classes also signed up for the reading classes?
c. What is the probability that a randomly chosen worker signed up for at least one of these two classes?
d. Are the events "signs up for the reading classes" and "signs up for the mathematics classes" statistically independent?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 50

A lawn-care service makes telephone solicitations, seeking customers for the coming season. A review of the records indicates that $15 \%$ of these solicitations produce new customers and that, of these new customers, $80 \%$ had used some rival service in the previous year. It is also estimated that, of all solicitation calls made, $60 \%$ are to people who had used a rival service the previous year. What is the probability that a call to a person who had used a rival service the previous year will produce a new customer for the lawncare service?

Sheryl Ezze

Numerade Educator

Problem 51

An editor may use all, some, or none of three possible strategies to enhance the sales of a book:
a. An expensive prepublication promotion
b. An expensive cover design
c. A bonus for sales representatives who meet predetermined sales levels
In the past, these three strategies have been applied simultaneously to only $2 \%$ of the company's books. Twenty percent of the books have had expensive cover designs, and, of these, $80 \%$ have had expensive prepublication promotion. A rival editor learns that a new book is to have both an expensive prepublication promotion and an expensive cover design and now wants to know how likely it is that a bonus scheme for sales representatives will be introduced. Compute the probability of interest to the rival editor.

Sheryl Ezze

Numerade Educator

Problem 52

Refer to Table 3.10.
What is the joint probability of "high income" and "never"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 53

Refer to Table 3.10.
What is the joint probability of "low income" and "regular"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 54

Refer to Table 3.10.
What is the joint probability of "middle income" and "never"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 55

Refer to Table 3.10.
What is the joint probability of "middle income" and "occasional"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 56

Refer to Table 3.10.
What is the conditional probability of "high income," given "never"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 57

Refer to Table 3.10.
What is the conditional probability of "low income," given "occasional"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 58

Refer to Table 3.10.
What is the conditional probability of "regular," given "high income"?
$$
\begin{aligned}
&\text { Table 3.10 Probabilities for Television Viewing and Income }\\
&\begin{array}{|lcccc|}
\hline \text { Viewing FrequenCY } & \text { High INCOMe } & \text { Middle INcome } & \text { Low Income } & \text { Totals } \\
\hline \text { Regular } & 0.10 & 0.15 & 0.05 & 0.30 \\
\text { Occasional } & 0.10 & 0.20 & 0.10 & 0.40 \\
\text { Never } & 0.05 & 0.05 & 0.20 & 0.30 \\
\text { Totals } & 0.25 & 0.40 & 0.35 & 1.00 \\
\hline
\end{array}
\end{aligned}
$$

Sheryl Ezze

Numerade Educator

Problem 59

The probability of a sale is 0.80 . What are the odds in favor of a sale?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 60

The probability of a sale is 0.50 . What are the odds in favor of a sale?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 61

Consider two groups of students: $B_1$, students who received high scores on tests, and $B_2$, students who received low scores on tests. In group $B_1, 80 \%$ study more than 25 hours per week, and in group $B_2, 40 \%$ study more than 25 hours per week. What is the overinvolvement ratio for high study levels in high test scores over low test scores?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 62

Consider two groups of students: $B_1$, students who received high scores on tests, and $B_2$, students who received low scores on tests. In group $B_1, 40 \%$ study more than 25 hours per week, and in group $B_2, 20 \%$ study more than 25 hours per week. What is the overinvolvement ratio for high study levels in high test scores over low test scores?

Sheryl Ezze

Numerade Educator

Problem 63

Consider two groups of students: $B_1$, students who received high scores on tests, and $B_2$, students who received low scores on tests. In group $B_1, 20 \%$ study more than 25 hours per week, and in group $B_2, 40 \%$ study more than 25 hours per week. What is the overinvolvement ratio for high study levels in high test scores over low test scores?

Sheryl Ezze

Numerade Educator

Problem 64

A survey carried out for a supermarket classified customers according to whether their visits to the store are frequent or infrequent and whether they often, sometimes, or never purchase generic products. The accompanying table gives the proportions of people surveyed in each of the six joint classifications.
$$
\begin{array}{|c|c|c|c|}
\hline {}{}{\begin{array}{l}
\text { Frequency of } \\
\text { Visit }
\end{array}} & {}{}{\begin{array}{c}
\text { Purchase of Generic } \\
\text { Products }
\end{array}} \\
\hline & \text { Often } & \text { Sometimes } & \text { Never } \\
\hline \text { Frequent } & 0.12 & 0.48 & 0.19 \\
\hline \text { Infrequent } & 0.07 & 0.06 & 0.08 \\
\hline
\end{array}
$$
a. What is the probability that a customer both is a frequent shopper and often purchases generic products?
b. What is the probability that a customer who never buys generic products visits the store frequently?
c. Are the events "never buys generic products" and "visits the store frequently" independent?
d. What is the probability that a customer who infrequently visits the store often buys generic products?
e. Are the events "often buys generic products" and "visits the store infrequently" independent?
f. What is the probability that a customer frequently visits the store?
g. What is the probability that a customer never buys generic products?
h. What is the probability that a customer either frequently visits the store or never buys generic products or both?

Sheryl Ezze

Numerade Educator

Problem 65

A consulting organization predicts whether corporations' earnings for the coming year will be unusually low, unusually high, or normal. Before deciding whether to continue purchasing these forecasts, a stockbroker compares past predictions with actual outcomes. The accompanying table shows proportions in the nine joint classifications.
$$
\begin{array}{lccc}
\hline & {}{}{\text { Prediction }} \\
& \text { Unusually } & \text { Normal } & \text { Unusually } \\
\hline \text { Outcome } & \text { High } & & \text { Low } \\
\text { Unusually high } & 0.23 & 0.12 & 0.03 \\
\text { Normal } & 0.06 & 0.22 & 0.08 \\
\text { Unusually low } & 0.01 & 0.06 & 0.19 \\
\hline
\end{array}
$$
a. What proportion of predictions have been for unusually high earnings?
b. What proportion of outcomes have been for unusually high earnings?
c. If a firm were to have unusually high earnings, what is the probability that the consulting organization would correctly predict this event?
d. If the organization predicted unusually high earnings for a corporation, what is the probability that these would materialize?
e. What is the probability that a corporation for which unusually high earnings had been predicted will have unusually low earnings?

Sheryl Ezze

Numerade Educator

Problem 66

Subscribers to a local newspaper were asked whether they regularly, occasionally, or never read the business section and also whether they had traded common stocks (or shares in a mutual fund) over the last year. The table shown here indicates the proportions of subscribers in six joint classifications.
$$
\begin{array}{lccc}
\hline {}{}{\begin{array}{l}
\text { Traded } \\
\text { Stocks }
\end{array}} & {}{}{\text { Read Business Section }} \\
& \text { Regularly } & \text { Occasionally } & \text { Never } \\
\hline \text { Yes } & 0.18 & 0.10 & 0.04 \\
\text { No } & 0.16 & 0.31 & 0.21 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen subscriber never reads the business section?
b. What is the probability that a randomly chosen subscriber has traded stocks over the last year?
c. What is the probability that a subscriber who never reads the business section has traded stocks over the last year?
d. What is the probability that a subscriber who traded stocks over the last year never reads the business section?
e. What is the probability that a subscriber who does not regularly read the business section traded stocks over the last year?

Sheryl Ezze

Numerade Educator

Problem 67

A corporation regularly takes deliveries of a particular sensitive part from three subcontractors. It found that the proportion of parts that are good or defective from the total received were as shown in the following table:
$$
\begin{array}{lccc}
\hline &{}{}{\text { Subcontractor }} \\
\text { Part } & \text { A } & \text { B } & \text { C } \\
\hline \text { Good } & 0.27 & 0.30 & 0.33 \\
\text { Defective } & 0.02 & 0.05 & 0.03 \\
\hline
\end{array}
$$
a. If a part is chosen randomly from all those received, what is the probability that it is defective?
b. If a part is chosen randomly from all those received, what is the probability it is from subcontractor $\mathrm{B}$ ?
c. What is the probability that a part from subcontractor $B$ is defective?
d. What is the probability that a randomly chosen defective part is from subcontractor $\mathrm{B}$ ?
e. Is the quality of a part independent of the source of supply?
f. In terms of quality, which of the three subcontractors is most reliable?

Sheryl Ezze

Numerade Educator

Problem 68

Students in a business statistics class were asked what grade they expected in the course and whether they worked on additional problems beyond those assigned by the instructor. The following table gives proportions of students in each of eight joint classifications.
$$
\begin{array}{lcccc}
\hline {}{}{\begin{array}{l}
\text { Worked } \\
\text { Problems }
\end{array}} & {}{}{\text { Expected Grade }} \\
& \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Below C } \\
\hline \text { Yes } & 0.12 & 0.06 & 0.12 & 0.02 \\
\text { No } & 0.13 & 0.21 & 0.26 & 0.08 \\
\hline
\end{array}
$$
a. Find the probability that a randomly chosen student from this class worked on additional problems.
b. Find the probability that a randomly chosen student from this class expects an A.
c. Find the probability that a randomly chosen student who worked on additional problems expects an A.
d. Find the probability that a randomly chosen student who expects an A worked on additional problems.
e. Find the probability that a randomly chosen student who worked on additional problems expects a grade below $B$.
f. Are "worked additional problems" and "expected grade" statistically independent?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 69

The accompanying table shows proportions of computer salespeople classified according to marital status and whether they left their jobs or stayed over a period of 1 year.
$$
\begin{array}{lcc}
\hline & {}{}{\text { Time on job }} \\
\text { Marital Status } & \geq \text { one year } & <\text { one year } \\
\hline \text { Married } & 0.64 & 0.13 \\
\text { Single } & 0.17 & 0.06 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen salesperson was married?
b. What is the probability that a randomly chosen salesperson left the job within the year?
c. What is the probability that a randomly chosen single salesperson left the job within the year?
d. What is the probability that a randomly chosen salesperson who stayed in the job over the year was married?

Sheryl Ezze

Numerade Educator

Problem 70

The accompanying table shows proportions of adults in metropolitan areas, categorized as to whether they are public-radio contributors and whether or not they voted in the last election.
$$
\begin{array}{lcc}
\hline \text { Voted } & \text { Contributors } & \text { Noncontributors } \\
\hline \text { Yes } & 0.63 & 0.13 \\
\text { No } & 0.14 & 0.10 \\
\hline
\end{array}
$$
a. What is the probability that a randomly chosen adult from this population voted?
b. What is the probability that a randomly chosen adult from this population contributes to public radio?
c. What is the probability that a randomly chosen adult from this population did not contribute and did not vote?

Sheryl Ezze

Numerade Educator

Problem 71

A campus student club distributed material about membership to new students attending an orientation meeting. Of those receiving this material $40 \%$ were men and $60 \%$ were women. Subsequently, it was found that $7 \%$ of the men and $9 \%$ of the women who received this material joined the club.
a. Find the probability that a randomly chosen new student who receives the membership material will join the club.
b. Find the probability that a randomly chosen new student who joins the club after receiving the membership material is a woman.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 72

An analyst attempting to predict a corporation's earnings next year believes that the corporation's business is quite sensitive to the level of interest rates. He believes that, if average rates in the next year are more than $1 \%$ higher than this year, the probability of significant earnings growth is 0.1 . If average rates next year are more than $1 \%$ lower than this year, the probability of significant earnings growth is estimated to be 0.8 . Finally, if average interest rates next year are within $1 \%$ of this year's rates, the probability for significant earnings growth is put at 0.5 . The analyst estimates that the probability is 0.25 that rates next year will be more than $1 \%$ higher than this year and 0.15 that they will be more than $1 \%$ lower than this year.
a. What is the estimated probability that both interest rates will be $1 \%$ higher and significant earnings growth will result?
b. What is the probability that this corporation will experience significant earnings growth?
c. If the corporation exhibits significant earnings growth, what is the probability that interest rates will have been more than $1 \%$ lower than in the current year?

Nick Johnson

Numerade Educator

Problem 73

Forty-two percent of a corporation's blue-collar employees were in favor of a modified health care plan, and $22 \%$ of its blue-collar employees favored a proposal to change the work schedule. Thirty-four percent of those favoring the health care plan modification favored the work schedule change.
a. What is the probability that a randomly selected blue-collar employee is in favor of both the modified health care plan and the changed work schedule?
b. What is the probability that a randomly chosen blue-collar employee is in favor of at least one of the two changes?
c. What is the probability that a blue-collar employee favoring the work schedule change also favors the modified health care plan?

Sheryl Ezze

Numerade Educator

Problem 74

The grades of a freshman college class, obtained after the first year of college, were analyzed. Seventy percent of the students in the top quarter of the college class had graduated in the upper $10 \%$ of their high school class, as had $50 \%$ of the students in the middle half of the college class and $20 \%$ of the students in the bottom quarter of the college class.
a. What is the probability that a randomly chosen freshman graduated in the upper $10 \%$ of his high school class?
b. What is the probability that a randomly chosen freshman who graduated in the upper $10 \%$ of the high school class will be in the top quarter of the college class?
c. What is the probability that a randomly chosen freshman who did not graduate in the upper $10 \%$ of the high school class will not be in the top quarter of the college class?

Sheryl Ezze

Numerade Educator

Problem 75

Before books aimed at preschool children are marketed, reactions are obtained from a panel of preschool children. These reactions are categorized as favorable, neutral, or unfavorable. Subsequently, book sales are categorized as high, moderate, or low, according to the norms of this market. Similar panels have evaluated 1,000 books in the past. The accompanying table shows their reactions and the resulting market performance of the books.
$$
\begin{array}{lccc}
\hline & {}{}{\text { Panel Reaction }} \\
\text { Sales } & \text { Favorable } & \text { Neutral } & \text { Unfavorable } \\
\hline \text { High } & 173 & 101 & 61 \\
\text { Moderate } & 88 & 211 & 70 \\
\text { Low } & 42 & 113 & 141 \\
\hline
\end{array}
$$
a. If the panel reaction is favorable, what is the probability that sales will be high?
b. If the panel reaction is unfavorable, what is the probability that sales will be low?
c. If the panel reaction is neutral or better, what is the probability that sales will be low?
d. If sales are low, what is the probability that the panel reaction was neutral or better?

Sheryl Ezze

Numerade Educator

Problem 76

A manufacturer produces boxes of candy, each containing 10 pieces. Two machines are used for this purpose. After a large batch has been produced, it is discovered that one of the machines, which produces $40 \%$ of the total output, has a fault that has led to the introduction of an impurity into $10 \%$ of the pieces of candy it makes. The other machine produced no defective pieces. From a single box of candy, one piece is selected at random and tested. If that piece contains no impurity, what is the probability that the faulty machine produced the box from which it came?

Sheryl Ezze

Numerade Educator

Problem 77

A student feels that $70 \%$ of her college courses have been enjoyable and the remainder have been boring. This student has access to student evaluations of professors and finds out that professors who had previously received strong positive evaluations from their students have taught $60 \%$ of his enjoyable courses and $25 \%$ of his boring courses. Next semester the student decides to take three courses, all from professors who have received strongly positive student evaluations. Assume that this student's reactions to the three courses are independent of one another.
a. What is the probability that this student will find all three courses enjoyable?
b. What is the probability that this student will find at least one of the courses enjoyable?

Nick Johnson

Numerade Educator

Problem 78

The following basic exercises use a sample space defined by events $A_1, A_2, B_1$, and $B_2$.
Given $P\left(A_1\right)=0.40, P\left(B_1 \mid A_1\right)=0.60$, and $P\left(B_1 \mid A_2\right)=0.70$, what is the probability of $P\left(A_1 \mid B_1\right)$ ?

Sheryl Ezze

Numerade Educator

Problem 79

The following basic exercises use a sample space defined by events $A_1, A_2, B_1$, and $B_2$.
Given $P\left(A_1\right)=0.80, P\left(B_1 \mid A_1\right)=0.60$, and $P\left(B_1 \mid A_2\right)=0.20$, what is the probability of $P\left(A_1 \mid B_1\right)$ ?

Sheryl Ezze

Numerade Educator

Problem 80

The following basic exercises use a sample space defined by events $A_1, A_2, B_1$, and $B_2$.
Given $P\left(A_1\right)=0.50, P\left(B_1 \mid A_1\right)=0.40$, and $P\left(B_1 \mid A_2\right)=0.70$, what is the probability of $P\left(A_1 \mid B_2\right)$ ?

Sheryl Ezze

Numerade Educator

Problem 81

The following basic exercises use a sample space defined by events $A_1, A_2, B_1$, and $B_2$.
Given $P\left(A_1\right)=0.40, P\left(B_1 \mid A_1\right)=0.60$, and $P\left(B_1 \mid A_2\right)=0.70$, what is the probability of $P\left(A_2 \mid B_2\right)$ ?

Sanchit Jain

Numerade Educator

Problem 82

The following basic exercises use a sample space defined by events $A_1, A_2, B_1$, and $B_2$.
Given $P\left(A_1\right)=0.60, P\left(B_1 \mid A_1\right)=0.60$, and $P\left(B_1 \mid A_2\right)=0.40$, what is the probability of $P\left(A_1 \mid B_1\right)$ ?

Sheryl Ezze

Numerade Educator

Problem 83

A publisher sends advertising materials for an accounting text to $80 \%$ of all professors teaching the appropriate accounting course. Thirty percent of the professors who received this material adopted the book, as did $10 \%$ of the professors who did not receive the material. What is the probability that a professor who adopts the book has received the advertising material?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 84

A stock market analyst examined the prospects of the shares of a large number of corporations. When the performance of these stocks was investigated one year later, it turned out that $25 \%$ performed much better than the market average, $25 \%$, much worse, and the remaining $50 \%$, about the same as the average. Forty percent of the stocks that turned out to do much better than the market were rated good buys by the analyst, as were $20 \%$ of those that did about as well as the market and $10 \%$ of those that did much worse. What is the probability that a stock rated a good buy by the analyst performed much better than the average?

Sheryl Ezze

Numerade Educator

Problem 85

The Watts New Lightbulb Corporation ships large consignments of lightbulbs to big industrial users. When the production process is functioning correctly, which is $90 \%$ of the time, $10 \%$ of all bulbs produced are defective. However, the process is susceptible to an occasional malfunction, leading to a defective rate of $50 \%$. If a defective bulb is found, what is the probability that the process is functioning correctly? If a nondefective bulb is found, what is the probability that the process is operating correctly?

Sheryl Ezze

Numerade Educator

Problem 86

You are the meat products manager for Gigantic Foods, a large retail supermarket food distributor who is studying the characteristics of its whole chicken product mix. Chickens are purchased from both Free Range Farms and Big Foods Ltd. Free Range Farms produces chickens that are fed with natural grains and grubs in open feeding areas. In their product mix, $10 \%$ of the processed chickens weigh less than 3 pounds. Big Foods Ltd. produces chickens in cages using enriched food grains for rapid growth. They note that $20 \%$ of their processed chickens weigh less than three poounds. Gigantic Foods purchases $40 \%$ of its chickens from Free Range Farms and mixes the products together with no identification of the supplier. Suppose you purchase a chicken that weighs more than three pounds. What is the probability the chicken came from Free Range Farms? If you purchase 5 chickens, what is the probability that at least 3 came from Free Range Farms?

Sheryl Ezze

Numerade Educator

Problem 87

You and a friend are big soccer fans and are debating the possibility that FC Barcelona will win the final of the UEFA Champions League against Manchester United. You are supporting Manchester United, but your friend tells you that the bookmakers have given the following odds for the game: 2:8 (Manchester United vs. FC Barcelona). What is the probability that Manchester United will win?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 88

Suppose that you have an intelligent friend who has not studied probability. How would you explain to your friend the distinction between mutually exclusive events and independent events? Illustrate your answer with suitable examples.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 89

State, with evidence, whether each of the following statements is true or false:
a. The complement of the union of two events is the intersection of their complements.
b. The sum of the probabilities of collectively exhaustive events must equal 1.
c. The number of combinations of $x$ objects chosen from $n$ is equal to the number of combinations of $(n-x)$ objects chosen from $n$, where $1 \leq x \leq(n-1)$.
d. If $A$ and $B$ are two events, the probability of $A$, given $B$, is the same as the probability of $B$, given $A$, if the probability of $A$ is the same as the probability of $B$.
e. If an event and its complement are equally likely to occur, the probability of that event must be 0.5 .
f. If $A$ and $B$ are independent, then $\bar{A}$ and $\bar{B}$ must be independent.
g. If $A$ and $B$ are mutually exclusive, then $\bar{A}$ and $\bar{B}$ must be mutually exclusive.

Sheryl Ezze

Numerade Educator

Problem 90

Explain carefully the meaning of conditional probability. Why is this concept important in discussing the chance of an event's occurrence?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 91

Bayes' theorem is important because it provides a rule for moving from a prior probability to a posterior probability. Elaborate on this statement so that it would be well understood by a fellow student who has not yet studied probability.

Hossam Mohamed

Numerade Educator

Problem 92

State, with evidence, whether each of the following statements is true or false:
a. The probability of the union of two events cannot be less than the probability of their intersection.
b. The probability of the union of two events cannot be more than the sum of their individual probabilities.
c. The probability of the intersection of two events cannot be greater than either of their individual probabilities.
d. An event and its complement are mutually exclusive.
e. The individual probabilities of a pair of events cannot sum to more than 1.
f. If two events are mutually exclusive, they must also be collectively exhaustive.
g. If two events are collectively exhaustive, they must also be mutually exclusive.

Sheryl Ezze

Numerade Educator

Problem 93

Distinguish among joint probability, marginal probability, and conditional probability. Provide some examples to make the distinctions clear.

Sheryl Ezze

Numerade Educator

Problem 94

State, with evidence, whether each of the following claims is true or false:
a. The conditional probability of $A$, given $B$, must be at least as large as the probability of $A$.
b. An event must be independent of its complement.
c. The probability of $A$, given $B$, must be at least as large as the probability of the intersection of $A$ and $B$.
d. The probability of the intersection of two events cannot exceed the product of their individual probabilities.
e. The posterior probability of any event must be at least as large as its prior probability.

Nick Johnson

Numerade Educator

Problem 95

Show that the probability of the union of events $A$ and $B$ can be written as follows:
$$
P(A \cup B)=P(A)+P(B)[1-P(A \mid B)]
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 96

An insurance company estimated that $30 \%$ of all automobile accidents were partly caused by weather conditions and that $20 \%$ of all automobile accidents involved bodily injury. Further, of those accidents that involved bodily injury, $40 \%$ were partly caused by weather conditions.
a. What is the probability that a randomly chosen accident both was partly caused by weather conditions and involved bodily injury?
b. Are the events "partly caused by weather conditions" and "involved bodily injury" independent?
c. If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury?
d. What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury?

Sheryl Ezze

Numerade Educator

Problem 97

A company places a rush order for wire of two thicknesses. Consignments of each thickness are to be sent immediately when they are available. Previous experience suggests that the probability is 0.8 that at least one of these consignments will arrive within a week. It is also estimated that, if the thinner wire arrives within a week, the probability is 0.4 that the thicker wire will also arrive within a week. Further, it is estimated that, if the thicker wire arrives within a week, the probability is 0.6 that the thinner wire will also arrive within a week.
a. What is the probability that the thicker wire will arrive within a week?
b. What is the probability that the thinner wire will arrive within a week?
c. What is the probability that both consignments will arrive within a week?

Amany Waheeb

Numerade Educator

Problem 98

Staff, Inc., a management consulting company, is surveying the personnel of Acme Ltd. It determined that $35 \%$ of the analysts have an MBA and that $40 \%$ of all analysts are over age 35 . Further, of those who have an MBA, $30 \%$ are over age 35 .
a. What is the probability that a randomly chosen analyst both has an MBA and also is over age 35 ?
b. What is the probability that a randomly chosen analyst who is over age 35 has an MBA?
c. What is the probability that a randomly chosen analyst has an MBA or is over age 35?
d. What is the probability that a randomly chosen analyst who is over age 35 does not have an MBA?
e. Are the events MBA and over age 35 independent?
f. Are the events MBA and over age 35 mutually exclusive?
g. Are the events MBA and over age 35 collectively exhaustive?

Sheryl Ezze

Numerade Educator

Problem 99

In a campus restaurant it was found that $35 \%$ of all customers order vegetarian meals and that $50 \%$ of all customers are students. Further, $25 \%$ of all customers who are students order vegetarian meals.
a. What is the probability that a randomly chosen customer both is a student and orders a vegetarian meal?
b. If a randomly chosen customer orders a vegetarian meal, what is the probability that the customer is a student?
c. What is the probability that a randomly chosen customer both does not order a vegetarian meal and is not a student?
d. Are the events "customer orders a vegetarian meal" and "customer is a student" independent?
e. Are the events "customer orders a vegetarian meal" and "customer is a student" mutually exclusive?
f. Are the events "customer orders a vegetarian meal" and "customer is a student" collectively exhaustive?

Sheryl Ezze

Numerade Educator

Problem 100

It is known that $20 \%$ of all farms in a state exceed 160 acres and that $60 \%$ of all farms in that state are owned by persons over 50 years old. Of all farms in the state exceeding 160 acres, $55 \%$ are owned by persons over 50 years old.
a. What is the probability that a randomly chosen farm in this state both exceeds 160 acres and is owned by a person over 50 years old?
b. What is the probability that a farm in this state either is bigger than 160 acres or is owned by a person over 50 years old (or both)?
c. What is the probability that a farm in this state, owned by a person over 50 years old, exceeds 160 acres?
d. Are size of farm and age of owner in this state statistically independent?

Sheryl Ezze

Numerade Educator

Problem 101

In a large corporation, $80 \%$ of the employees are men and $20 \%$ are women. The highest levels of education obtained by the employees are graduate training for $10 \%$ of the men, undergraduate training for $30 \%$ of the men, and high school training for $60 \%$ of the men. The highest levels of education obtained are also graduate training for $15 \%$ of the women, undergraduate training for $40 \%$ of the women, and high school training for $45 \%$ of the women.
a. What is the probability that a randomly chosen employee will be a man with only a high school education?
b. What is the probability that a randomly chosen employee will have graduate training?
c. What is the probability that a randomly chosen employee who has graduate training is a man?
d. Are gender and level of education of employees in this corporation statistically independent?
e. What is the probability that a randomly chosen employee who has not had graduate training is a woman?

Sheryl Ezze

Numerade Educator

Problem 102

A large corporation organized a ballot for all its workers on a new bonus plan. It was found that $65 \%$ of all night-shift workers favored the plan and that $40 \%$ of all female workers favored the plan. Also, $50 \%$ of all employees are night-shift workers and $30 \%$ of all employees are women. Finally, $20 \%$ of all night-shift workers are women.
a. What is the probability that a randomly chosen employee is a woman in favor of the plan?
b. What is the probability that a randomly chosen employee is either a woman or a night-shift worker (or both)?
c. Is employee gender independent of whether the night shift is worked?
d. What is the probability that a female employee is a night-shift worker?
e. If $50 \%$ of all male employees favor the plan, what is the probability that a randomly chosen employee both does not work the night shift and does not favor the plan?

Sheryl Ezze

Numerade Educator

Problem 103

A jury of 12 members is to be selected from a panel consisting of 8 men and 8 women.
a. How many different jury selections are possible?
b. If the choice is made randomly, what is the probability that a majority of the jury members will be men?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 104

A consignment of 12 electronic components contains 1 component that is faulty. Two components are chosen randomly from this consignment for testing.
a. How many different combinations of 2 components could be chosen?
b. What is the probability that the faulty component will be chosen for testing?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 105

Tiger Funds Ltd. operates a number of mutual funds in high technology and in financial sectors. Hussein Roberts is a fund manager who runs a major fund that includes a wide variety of technology stocks. As fund manager he decides which stocks should be purchased for the mutual fund. The compensation plan for fund managers includes a first-year bonus for each stock purchased by the manager that gains more than $10 \%$ in the first six months it is held. Of those stocks that the company holds, $40 \%$ are up in value after being held for two years. In reviewing the performance of Mr. Roberts, they found that he received a first-year bonus for $60 \%$ of the stocks that he purchased that were up after two years. He also received a first-year bonus for $40 \%$ of the stocks he purchased that were not up after two years. What is the probability that a stock will be up after two years given that $\mathrm{Mr}$. Roberts received a first-year bonus?

Check back soon!

Problem 106

Of 100 patients with a certain disease, 10 were chosen at random to undergo a drug treatment that increases the cure rate from $50 \%$ for those not given the treatment to $75 \%$ for those given the drug treatment.
a. What is the probability that a randomly chosen patient both was cured and was given the drug treatment?
b. What is the probability that a patient who was cured had been given the drug treatment?
c. What is the probability that a specific group of 10 patients was chosen to undergo the drug treatment? (Leave your answer in terms of factorials.)

Sheryl Ezze

Numerade Educator

Problem 107

Subscriptions to a particular magazine are classified as gift, previous renewal, direct mail, and subscription service. In January $8 \%$ of expiring subscriptions were gifts; $41 \%$, previous renewal; $6 \%$, direct mail; and $45 \%$, subscription service. The percentages of renewals in these four categories were $81 \%, 79 \%, 60 \%$, and $21 \%$, respectively. In February of the same year, $10 \%$ of expiring subscriptions were gift; $57 \%$, previous renewal; $24 \%$, direct mail; and $9 \%$, subscription service. The percentages of renewals were $80 \%, 76 \%, 51 \%$, and $14 \%$, respectively.
a. Find the probability that a randomly chosen subscription expiring in January was renewed.
b. Find the probability that a randomly chosen subscription expiring in February was renewed.
c. Verify that the probability in part (b) that is higher than that in part (a). Do you believe that the editors of this magazine should view the change from January to February as a positive or negative development?

Jason Gerber

Numerade Educator

Problem 108

The Customs Inspection agency at international airports has developed a traveler profiling system (TPS) to detect passengers who are trying to bring more liquor into the country than is allowed by present regulations. Long-term studies indicate that $20 \%$ of the passengers are carrying more liquor than is allowed. Tests on the new TPS scheme has shown that of those carrying illegal amounts of liquor, $80 \%$ will be identified and subject to complete luggage search. In addition 20\% of those not carrying illegal amounts of liquor will also be identified by TPS and subject to a complete luggage search.
If a passenger is identified by TPS, what is the probability that the passenger is carrying an illegal amount of liquor? Comment on the value of this system.

Sheryl Ezze

Numerade Educator

Problem 109

3.109 In a large city, $8 \%$ of the inhabitants have contracted a particular disease. A test for this disease is positive in $80 \%$ of people who have the disease and is negative in $80 \%$ of people who do not have the disease. What is the probability that a person for whom the test result is positive has the disease?

Sheryl Ezze

Numerade Educator

Problem 110

A life insurance salesman finds that, of all the sales he makes, $70 \%$ are to people who already own policies. He also finds that, of all contacts for which no sale is made, $50 \%$ already own life insurance policies. Furthermore, $40 \%$ of all contacts result in sales. What is the probability that a sale will be made to a contact who already owns a policy?

Sheryl Ezze

Numerade Educator

Problem 111

A professor finds that she awards a final grade of A to $20 \%$ of her students. Of those who obtain a final grade of A, $70 \%$ obtained an A on the midterm examination. Also, $10 \%$ of the students who failed to obtain a final grade of A earned an A on the midterm exam. What is the probability that a student with an A on the midterm examination will obtain a final grade of A?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 112

The accompanying table shows, for 1,000 forecasts of earnings per share made by financial analysts, the numbers of forecasts and outcomes in particular categories (compared with the previous year).
$$
\begin{array}{lcrc}
\hline &{}{}{\text { Forecast }} \\
\text { Outcome } & \text { Improvement } & \text { About the } \\
\text { Same } & \text { Worse } \\
\hline \text { Improvement } & 210 & 82 & 66 \\
\text { About the same } & 106 & 153 & 75 \\
\text { Worse } & 75 & 84 & 149 \\
\hline
\end{array}
$$
a. Find the probability that if the forecast is for a worse performance in earnings, this outcome will result.
b. If the forecast is for an improvement in earnings, find the probability that this outcome fails to result.

Sheryl Ezze

Numerade Educator

Problem 113

A dean has found that $62 \%$ of entering freshmen and $78 \%$ of community college transfers eventually graduate. Of all entering students, $73 \%$ are entering freshmen and the remainder are community college transfers.
a. What is the probability that a randomly chosen entering student is an entering freshman who will eventually graduate?
b. Find the probability that a randomly chosen entering student will eventually graduate.
c. What is the probability that a randomly chosen entering student either is an entering freshman or will eventually graduate (or both)?
d. Are the events "eventually graduates" and "enters as community college transfer" statistically independent?

Sheryl Ezze

Numerade Educator

Problem 114

A market-research group specializes in providing assessments of the prospects of sites for new children's toy stores in shopping centers. The group assesses prospects as good, fair, or poor. The records of assessments made by this group were examined, and it was found that for all stores that had annual sales over $$\$ 1,000,000$$, the assessments were good for $70 \%$, fair for $20 \%$, and poor for $10 \%$. For all stores that turned out to be unsuccessful, the assessments were good for $20 \%$, fair for $30 \%$, and poor for $50 \%$. It is known that $60 \%$ of new clothing stores are successful and $40 \%$ are unsuccessful.
a. For a randomly chosen store, what is the probability that prospects will be assessed as good?
b. If prospects for a store are assessed as good, what is the probability that it will be successful?
c. Are the events "prospects assessed as good" and "store is successful" statistically independent?
d. Suppose that five stores are chosen at random. What is the probability that at least one of them will be successful?

Sheryl Ezze

Numerade Educator

Problem 115

A restaurant manager classifies customers as regular, occasional, or new, and finds that of all customers $50 \%, 40 \%$, and $10 \%$, respectively, fall into these categories. The manager found that wine was ordered by $70 \%$ of the regular customers, by $50 \%$ of the occasional customers, and by $30 \%$ of the new customers.
a. What is the probability that a randomly chosen customer orders wine?
b. If wine is ordered, what is the probability that the person ordering is a regular customer?
c. If wine is ordered, what is the probability that the person ordering is an occasional customer?

Sheryl Ezze

Numerade Educator

Problem 116

A record-store owner assesses customers entering the store as high school age, college age, or older, and finds that of all customers $30 \%, 50 \%$, and $20 \%$, respectively, fall into these categories. The owner also found that purchases were made by $20 \%$ of high school age customers, by $60 \%$ of college age customers, and by $80 \%$ of older customers.
a. What is the probability that a randomly chosen customer entering the store will make a purchase?
b. If a randomly chosen customer makes a purchase, what is the probability that this customer is high school age?

Sheryl Ezze

Numerade Educator

Problem 117

Note that this exercise represents a completely imaginary situation. Suppose that a statistics class contained exactly 8 men and 8 women. You have discovered that the teacher decided to assign $5 \mathrm{Fs}$ on an exam by randomly selecting names from a hat. He concluded that this would be easier than actually grading all those papers and that his students are all equally skilled in statistics-but someone has to get an F. What is the probability that all 5 Fs were given to male students?

Sheryl Ezze

Numerade Educator

Problem 118

A survey on the best Asian tourist destinations showed that, out of 70 people, 23 ranked Singapore as first, whereas 15 put Hong Kong in first place, 11 put Shanghai first, 7 put Beijing first, and the rest of them chose Tokyo. On the basis of this data, calculate the following.
a. The probability of the preferred destination being a city in China. (In this specific case, Hong Kong is not considered part of China.)
b. The probability of the preferred destination not being a Chinese city. (In this case, Hong Kong is considered a Chinese city, even if outside China.)
c. The probability of the preferred destination being Tokyo.
d. The probability of the preferred destination not being Singapore.

Sheryl Ezze

Numerade Educator

Problem 119

You are responsible for detecting the source of the error when a computer system fails. From your analysis you know that the source of error is the disk drive, the computer memory, or the operating system. You know that $50 \%$ of the errors are disk drive errors, $30 \%$ are computer memory errors, and the remainder are operating system errors. From the component performance standards, you know that when a disk drive error occurs, the probability of failure is 0.60 ; when a computer memory error occurs, the probability of failure is 0.7; and when an operating system error occurs, the probability of failure is 0.3 . Given the information from the component performance standards, what is the probability of a disk drive error, given that a failure occurred?

Sheryl Ezze

Numerade Educator

Problem 120

After meeting with the regional sales managers, Lauretta Anderson, president of Cowpie Computers, Inc., you find that she believes that the probability that sales will grow by $10 \%$ in the next year is 0.70 . After coming to this conclusion, she receives a report that John Cadariu of Minihard Software, Inc., has just announced a new operating system that will be available for customers in 8 months. From past history she knows that in situations where growth has eventually occurred, new operating systems have been announced $30 \%$ of the time. However, in situations where growth has not eventually occurred, new operating systems have been announced $10 \%$ of the time. Based on all these facts, what is the probability that sales will grow by $10 \%$ ?

Nick Johnson

Numerade Educator

Problem 121

Sally Firefly purchases hardwood lumber for a custom furniture-building shop. She uses three suppliers, Northern Hardwoods, Mountain Top, and Spring Valley. Lumber is classified as either clear or has defects, which includes $20 \%$ of the pile. A recent analysis of the defect lumber pile showed that $30 \%$ came from Northern Hardwoods and $50 \%$ came from Mountain Top. Analysis of the clear pile indicates that $40 \%$ came from Northern and $40 \%$ came from Spring Valley. What is the percent of clear lumber from each of the three suppliers? What is the percent of lumber from each of the three suppliers?

Nick Johnson

Numerade Educator

Problem 122

Robert Smith uses either regular plowing or minimal plowing to prepare the cornfields on his Minnesota farm. Regular plowing was used for $40 \%$ of the field acreage. Analysis after the crop was harvested showed that $50 \%$ of the high-yield acres were from minimalplowing fields and $40 \%$ of the low yield fields were from fields with regular plowing. What is the probability of a high yield if regular plowing is used? What is the probability that a field with high yield had been prepared using regular plowing?

Nick Johnson

Numerade Educator