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Statistics for business and economics

David R. Anderson et al.

Chapter 4

Introduction to Probability - all with Video Answers

Educators

Chapter Questions

01:42

Problem 1

An experiment has three steps with three outcomes possible for the first step, two outcomes possible for the second step and four outcomes possible for the third step. How many experimental outcomes exist for the entire experiment?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
01:51

Problem 2

How many ways can three items be selected from a group of six items? Use the letters A, B, C, D, E and $\mathrm{F}$ to identify the items, and list each of the different combinations of three items.

Nick Johnson
Nick Johnson
Numerade Educator
04:26

Problem 3

How many permutations of three items can be selected from a group of six? Use the letters A, B, C, $\mathrm{D}, \mathrm{E}$ and $\mathrm{F}$ to identify the items, and list each of the permutations of items $\mathrm{B}, \mathrm{D}$ and $\mathrm{F}$.

Sandra Kudolo
Sandra Kudolo
Numerade Educator
02:49

Problem 4

Consider the experiment of tossing a coin three times.
a. Develop a tree diagram for the experiment.
b. List the experimental outcomes.
c. What is the probability for each experimental outcome?

SY
Song Yh
Numerade Educator
01:59

Problem 5

Suppose an experiment has five equally likely outcomes: $E_1, E_2, E_3, E_4, E_5$. Assign probabilities to each outcome and show that the requirements in equations (4.3) and (4.4) are satisfied. What method did you use?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
01:24

Problem 6

An experiment with three outcomes has been repeated 50 times, and it was learned that $E_1$ occurred 20 times, $E_2$ occurred 13 times and $E_3$ occurred 17 times. Assign probabilities to the outcomes. What method did you use?

SY
Song Yh
Numerade Educator
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Problem 7

A decision-maker subjectively assigned the following probabilities to the four outcomes of an experiment: $P\left(E_1\right)=0.10, P\left(E_2\right)=0.15, P\left(E_3\right)=0.40$ and $P\left(E_4\right)=0.20$. Are these probability assignments valid? Explain.

James Kiss
James Kiss
Numerade Educator
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Problem 8

Applications for zoning changes in a large metropolitan city go through a two-step process: a review by the planning commission and a final decision by the city council. At step 1 the planning commission reviews the zoning change request and makes a positive or negative recommendation concerning the change. At step 2 the city council reviews the planning commission's recommendation and then votes to approve or to disapprove the zoning change. Suppose the developer of an apartment complex submits an application for a zoning change. Consider the application process as an experiment.
a. How many sample points are there for this experiment? List the sample points.
b. Construct a tree diagram for the experiment.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:13

Problem 9

A total of 11 Management students, four International Management and American Business Studies (IMABS) and eight International Management and French Studies (IMF) students have volunteered to take part in an inter-university toumament.
a. How many different ways can a team consisting of eight Management students, two IMABS and five IMF students be selected?
b. If after the team has been selected, one Management, one IMABS and two IMF students are found to be suffering from glandular fever and are unable to play, what is the probability that the team will not have to be changed?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:34

Problem 10

A company that franchises coffee houses conducted taste tests for a new coffee product. Four blends were prepared, then randomly chosen individuals were asked to taste the blends and state which one they liked best. Results of the taste test for 100 individuals are given.$$
\begin{array}{cc}
\text { Blend } & \text { Number choosing } \\
\hline 1 & 20 \\
2 & 30 \\
3 & 35 \\
4 & 15
\end{array}
$$a. Define the experiment being conducted. How many times was it repeated?
b. Prior to conducting the experiment, it is reasonable to assume preferences for the four blends are equal. What probabilities would you assign to the experimental outcomes prior to conducting the taste test? What method did you use?
c. After conducting the taste test, what probabilities would you assign to the experimental outcomes? What method did you use?

Kari Hasz
Kari Hasz
Numerade Educator
03:56

Problem 11

A company that manufactures toothpaste is studying five different package designs. Assuming that one design is just as likely to be selected by a consumer as any other design, what selection probability would you assign to each of the package designs? In an actual experiment, 100 consumers were asked to pick the design they preferred. The following data were obtained. Do the data confirm the belief that one design is just as likely to be selected as another? Explain.$$
\begin{array}{cc}
\text { Design times } & \text { Number of preferred } \\
\hline 1 & 5 \\
2 & 15 \\
3 & 30 \\
4 & 40 \\
5 & 10
\end{array}
$$

Sandra Kudolo
Sandra Kudolo
Numerade Educator
02:06

Problem 12

An experiment has four equally likely outcomes: $E_1, E_2, E_3$ and $E_4$.
a. What is the probability that $E_2$ occurs?
b. What is the probability that any two of the outcomes occur (e.g. $E_1$ or $E_3$ )?
c. What is the probability that any three of the outcomes occur (e.g. $E_1$ or $E_2$ or $\left.E_4\right)$ ?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
04:27

Problem 13

Consider the experiment of selecting a playing card from a deck of 52 playing cards. Each card corresponds to a sample point with a $1 / 52$ probability.
a. List the sample points in the event an ace is selected.
b. List the sample points in the event a club is selected.
c. List the sample points in the event a face card (jack, queen or king) is selected.
d. Find the probabilities associated with each of the events in parts (a), (b) and (c).

Sandra Kudolo
Sandra Kudolo
Numerade Educator
07:05

Problem 14

Consider the experiment of rolling a pair of dice. Suppose that we are interested in the sum of the face values showing on the dice.
a. How many sample points are possible? (Hint: Use the counting rule for multiple-step experiments.)
b. List the sample points.
c. What is the probability of obtaining a value of 7 ?
d. What is the probability of obtaining a value of 9 or greater?
e. Because each roll has six possible even values $(2,4,6,8,10$ and 12$)$ and only five possible odd values $(3,5,7,9$ and 11$)$, the dice should show even values more often than odd values. Do you agree with this statement? Explain.
f. What method did you use to assign the probabilities requested?

Pratyush Raitan
Pratyush Raitan
Numerade Educator
04:43

Problem 15

Refer to the KPL sample points and sample point probabilities in Tables 4.2 and 4.3.
a. The design stage (stage 1) will run over budget if it takes four months to complete. List the sample points in the event the design stage is over budget.
b. What is the probability that the design stage is over budget?
c. The construction stage (stage 2 ) will run over budget if it takes eight months to complete. List the sample points in the event the construction stage is over budget.
d. What is the probability that the construction stage is over budget?
e. What is the probability that both stages are over budget?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
01:43

Problem 16

Suppose that a manager of a large apartment complex provides the following subjective probability estimates about the number of vacancies that will exist next month.$$
\begin{array}{lc}
\text { Vacancies } & \text { Probability } \\
\hline 0 & 0.10 \\
1 & 0.15 \\
2 & 0.30 \\
3 & 0.20 \\
4 & 0.15 \\
5 & 0.10
\end{array}
$$Provide the probability of each of the following events.
a. No vacancies.
b. At least four vacancies.
c. Two or fewer vacancies.

Sandra Kudolo
Sandra Kudolo
Numerade Educator
02:36

Problem 17

When three marksmen take part in a shooting contest, their chances of hitting the target are $1 / 2$, $1 / 3$ and $1 / 4$ respectively. If all three marksmen fire at it simultaneously
a. What is the chance that one and only one bullet will hit the target?
b. What is the chance that two marksmen will hit the target (and therefore one will not)?
c. What is the chance that all three marksmen will hit the target?

Aman Gupta
Aman Gupta
Numerade Educator
07:13

Problem 18

Suppose that we have a sample space with five equally likely experimental outcomes: $E_1, E_2, E_3$, $E_4, E_5$. Let:
$$
\begin{aligned}
& A=\left\{E_1, E_2\right\} \\
& B=\left\{E_3, E_4\right\} \\
& C=\left\{E_2, E_3, E_5\right\}
\end{aligned}
$$
a. Find $P(A), P(B)$ and $P(C)$.
b. Find $P(A \cup B)$. Are $A$ and $B$ mutually exclusive?
c. Find $\bar{A}, \bar{C}, P(\bar{A})$ and $P(\bar{C})$.
d. Find $A \cup \bar{B}$ and $P(A \cup \bar{B})$.
e. Find $P(B \cup C)$.

Sandra Kudolo
Sandra Kudolo
Numerade Educator
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Problem 19

Suppose that we have a sample space $S=\left\{E_1, E_2, E_3, E_4, E_5, E_6, E_7\right\}$, where $E_1, E_2, \ldots, E_7$ denote the sample points. The following probability assignments apply: $P\left(E_1\right)=0.05, P$ $\left(E_2\right)=0.20, P\left(E_3\right)=0.20, P\left(E_4\right)=0.25, P\left(E_5\right)=0.15, P\left(E_6\right)=0.10$, and $P\left(E_7\right)=0.05$. Let:
$$
\begin{aligned}
& A=\left\{E_1, E_2\right\} \\
& B=\left\{E_3, E_4\right] \\
& C=\left\{E_2, E_3, E_5\right\}
\end{aligned}
$$
a. Find $P(A), P(B)$, and $P(C)$.
b. Find $A \cup B$ and $P(A \cup B)$.
c. Find $A \cap B$ and $P(A \cap B)$.
d. Are events $A$ and $C$ mutually exclusive?
e. Find $\bar{B}$ and $P(\bar{B})$.

Danielle Fairburn
Danielle Fairburn
Numerade Educator
03:11

Problem 20

A survey of magazine subscribers showed that 45.8 per cent rented a car during the past 12 months for business reasons, 54 per cent rented a car during the past 12 months for personal reasons and 30 per cent rented a car during the past 12 months for both business and personal reasons.
a. What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons?
b. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
04:20

Problem 21

Suppose that we have two events, $A$ and $B$, with $P(A)=0.50, P(B)=0.60$ and $P(A \cap B)=0.40$.
a. Find $P(A \mid B)$.
b. Find $P(B \mid A)$.
c. Are $A$ and $B$ independent? Why or why not?

MA
Muhammad Awais
Numerade Educator
05:36

Problem 22

Assume that we have two events, $A$ and $B$, that are mutually exclusive. Assume further that we know $P(A)=0.30$ and $P(B)=0.40$.
a. What is $P(A \cap B)$ ?
b. What is $P(A \mid B)$ ?
c. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Do you agree with this statement? Use the probability information in this problem to justify your answer.
d. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
00:50

Problem 23

A Paris nightclub obtains the following data on the age and marital status of 140 customers.
$$
\begin{array}{lcc}
& {\text { Marital status }} \\
{ 2 - 3 } \text { Age } & \text { Single } & \text { Married } \\
\hline \text { Under 30 } & 77 & 14 \\
\text { 30 or over } & 28 & 21
\end{array}
$$
a. Develop a joint probability table for these data.
b. Use the marginal probabilities to comment on the age of customers attending the club.
c. Use the marginal probabilities to comment on the marital status of customers attending the club.
d. What is the probability of finding a customer who is single and under the age of 30 ?
e. If a customer is under 30 , what is the probability that he or she is single?
f. Is marital status independent of age? Explain, using probabilities.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
09:02

Problem 24

A slot machine in Melbourne has a hold facility. A gambler experiments with this to see if their success rate is higher when they use 'hold' compared to when they do not.
The results from 120 plays can be summarized as follows.
$$
\begin{array}{lcc}
& \text { Win } & \text { Lose } \\
\hline \text { Hold } & 14 & 36 \\
\text { Not hold } & 10 & 60
\end{array}
$$
What is the probability that the gambler:
a. Holds?
b. Wins?
c. Wins given that they held?
d. Held and lost?
e. Held given that they won?

Heena Haldankar
Heena Haldankar
Numerade Educator
02:05

Problem 25

A sample of convictions and compensation orders issued at a number of Manx courts was followed up to see whether the offender had paid the compensation to the victim. Details by gender of offender are as follows:
$$
\begin{array}{lccc}
& {\text { Payment outcome }} \\
{ 2 - 4 } \begin{array}{l}
\text { Offender }
\end{array} & \text { Paid in full } & \text { Part paid } & \text { Nothing paid } \\
\hline \text { Male } & 754 & 62 & 61 \\
\text { Female } & 157 & 7 & 6
\end{array}
$$
a. What is the probability that no compensation was paid?
b. What is the probability that the offender was not male given that compensation was part paid?

Christopher Stanley
Christopher Stanley
Numerade Educator
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Problem 26

A purchasing agent in Haifa placed rush orders for a particular raw material with two different suppliers, $A$ and $B$. If neither order arrives in four days, the production process must be shut down until at least one of the orders arrives. The probability that supplier $A$ can deliver the material in four days is 0.55 . The probability that supplier $B$ can deliver the material in four days is 0.35 .
a. What is the probability that both suppliers will deliver the material in four days? Because two separate suppliers are involved, we are willing to assume independence.
b. What is the probability that at least one supplier will deliver the material in four days?
c. What is the probability that the production process will be shut down in four days because of a shortage of raw material (that is, both orders are late)?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
06:32

Problem 27

The prior probabilities for events $A_1$ and $A_2$ are $P\left(A_1\right)=0.40$ and $P\left(A_2\right)=0.60$. It is also known that $P\left(A_1 \cap A_2\right)=0$. Suppose $P\left(B \mid A_1\right)=0.20$ and $P\left(B \mid A_2\right)=0.05$.
a. Are $A_1$ and $A_2$ mutually exclusive? Explain.
b. Compute $P\left(A_1 \cap B\right)$ and $P\left(A_2 \cap B\right)$.
c. Compute $P(B)$.
d. Apply Bayes' theorem to compute $P\left(A_1 \mid B\right)$ and $P\left(A_2 \mid B\right)$.

Sandra Kudolo
Sandra Kudolo
Numerade Educator
04:07

Problem 28

The prior probabilities for events $A_1, A_2$ and $A_3$ are $P\left(A_1\right)=0.20, P\left(A_2\right)=0.50$ and $P\left(A_3\right)=0.30$. The conditional probabilities of event $B$ given $A_1, A_2$ and $A_3$ are $P\left(B \mid A_1\right)=0.50, P\left(B \mid A_2\right)=0.40$ and $P\left(B \mid A_3\right)=0.30$.
a. Compute $P\left(B \cap A_1\right), P\left(B \cap A_2\right)$ and $P\left(B \cap A_3\right)$.
b. Apply Bayes' theorem, equation (4.19), to compute the posterior probability $P\left(A_2 \mid B\right)$.
c. Use the tabular approach to applying Bayes' theorem to compute $P\left(A_1 \mid B\right), P\left(A_2 \mid B\right)$ and $P\left(A_3 \mid B\right)$.

Aman Gupta
Aman Gupta
Numerade Educator
01:32

Problem 29

Records show that for every 100 items produced in a factory during the day shift, two are defective and for every 100 items produced during the night shift, four are defective. What is the prior probability of the bid being successful (that is, prior to the request for additional information)?
a. If during a 24-hour period, 2000 items are produced during the day and 800 at night, what is the probability that an item picked at random from the output over 24 hours came from the night shift if it was defective?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:31

Problem 30

A company is about to sell to a new client. It knows from past experience that there is a real possibility that the client may default on payment. As a precaution the company checks with a consultant on the likelihood of the client defaulting in this case and is given an estimate of 20 per cent. Sometimes the consultant gets it wrong. Your own experience of the consultant is that he is correct 70 per cent of the time when he predicts that the client will default but that 20 per cent of clients who he believes will not default actually do.
a. What is the probability that the new client will not default?

Nick Johnson
Nick Johnson
Numerade Educator
00:54

Problem 31

In 2011, there were 1901 fatalities recorded on Britain's roads, 60 of which were for children (Department of Transport, 2012). Correspondingly, serious injuries totalled 23122 of which 20770 were for adults.
a. What is the probability of a serious injury given the victim was a child?
b. What is the probability that the victim was an adult given a fatality occurred?

Joshua Sieverding
Joshua Sieverding
Numerade Educator
02:19

Problem 32

The following cross-tabulation shows industry type and price/earnings $(P / E)$ ratio for 100 companies in the consumer products and banking industries.
$$
\begin{array}{lcccccr}
\hline {P / \text { E ratio }} \\
\hline \text { Industry } & 5-9 & 10-14 & 15-19 & 20-24 & 25-29 & \text { Total } \\
\hline \text { Consumer } & 4 & 10 & 18 & 10 & 8 & 50 \\
\text { Banking } & 14 & 14 & 12 & 6 & 4 & 50 \\
\text { Total } & 18 & 24 & 30 & 16 & 12 & 100
\end{array}
$$
a. What is the probability that a company had a $\mathrm{P} / \mathrm{E}$ greater than 9 and belonged to the consumer industry?
b. What is the probability that a company with a $P / E$ in the range $15-19$ belonged to the banking industry?

Dominador Tan
Dominador Tan
Numerade Educator
01:09

Problem 33

A large investment advisory service has a number of analysts who prepare detailed studies of individual companies. On the basis of these studies the analysts make 'buy' or 'sell' recommendations on the companies' shares. The company classes an excellent analyst as one who will be correct 80 per cent of the time, a good analyst as who will be correct 60 per cent of the time and a poor analyst who will be correct 40 per cent of the time.
Two years ago, the advisory service hired Mr Smith who came with considerable experience from the research department of another firm. At the time he was hired it was thought that the probability was 0.90 that he was an excellent analyst, 0.09 that he was a good analyst and 0.01 that he was a poor analyst. In the past two years he has made ten recommendations of which only three have been correct.
Assuming that each recommendation is an independent event what probability would you assign to Mr Smith being:
a. An excellent analyst?
b. A good analyst?
c. A poor analyst?

Lucas Finney
Lucas Finney
Numerade Educator
02:29

Problem 34

An electronic component is produced by four production lines in a manufacturing operation. The components are costly, are quite reliable and are shipped to suppliers in 50-component lots. Because testing is destructive, most buyers of the components test only a small number before deciding to accept or reject lots of incoming components. All four production lines usually only produce 1 per cent defective components which are randomly dispersed in the output. Unfortunately, production line 1 suffered mechanical difficulty and produced 10 per cent defectives during the month of April. This situation became known to the manufacturer after the components had been shipped. A customer received a lot in April and tested five components. Two failed. What is the probability that this lot came from production line 1 ?

James Kiss
James Kiss
Numerade Educator