solutions for exercise 1

8/3/2019 Solutions for Exercise 1

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THE UNIVERSITY OF DODOMA

SCHOOL OF SOCIAL SCIENCES

MS 200: QUANTITATIVE TECHNIQUES FOR BUSINESS DECISIOS

SUGGESTED SOLUTIONS FOR EXERCISE 1

Question 1

There are six people qualified for selection to the position of deputy mayor of Dar es Salaam City. Theoffice is to be filled by randomly selecting one person from the following list:

Person Sex Experience (Years)

1 Male 8

2 Female 7

3 Male 12

4 Female 4

5 Female 86 Male 4

Required:

Find the probability that the individual selected will be either a female or a person with more than 6

years of experience.

Solution

Let A be the event that the individual selected from the list is a female

Let B be the event that the individual selected from the list is a person with more than 6 years ofexperience.

Required: To determine P(AB)

But P(AB) = P(A) + P(B) - P(AB), since this is a case of non-mutually exclusive events.

Where;

P(A) = 3/6

P(B) = 4/6

P(AB) = 2/6

Therefore, P(AB) = 3/6 + 4/6 - 2/6

= 5/6.

Question 2

A financial analyst believes that if interest rates decrease in a given period, then the probability that the

stock market will go up is 0.80. The analyst further believes that interest have a 0.40 chance of

decreasing during a period in question. Given the above information, what is the probability that themarket will go up and interest rates will go down during the period in question?

1


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Solution

Let A be the event that the stock market goes up during a given period.

Let B be the event that interest rates decrease during the same period.

Required: To find P(AB)

Where, P(AB) = P(A/B) P(B)

From the information provided, we have:

P(A/B) = 0.80

P(B) = 0.40.

Thus, P(AB) = P(A/B) P(B)

= 0.80 0.40= 0.32.

Question 3

b) The probability that MICRONIX SYSTEMS salesperson sells a computer to a prospectivecustomer on the first visit is 0.4. If the salesperson fails to make the sale on the first visit, the

probability that the sale will be made on the second visit is 0.65. The sales person never visits a

prospective customer more than twice. Find the probability that the salesperson will make a sale to a

particular customer.

Solution to part (b)

Let A be the event that MICRONIX SYSTEMS sales person sells the computer to a prospective

customer on the first visit.Let B be the event that MICRONIX SYSTEMS sales person sells the computer to a prospective

customer on the second visit.

Required: To find

Because the sale will only be made on the second visit if it has not taken place on the first.

From the information provided, we have:

2

)()/()()()( APABPAPBAPAP +=+

65.0)/(

6.0)(

4.0)(

=

=

=

ABP

AP

AP

.79.0

60.065.040.0)()(,

=

+=+ BAPAPTherefore


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Question 4

The figure below shows a schematic representation of a system comprised of three components.

A system Comprised of Three Components in Series

Input Output

The system operates properly only if at least two of its components operate properly. The three

components are said to operate in series. The components could represent the functions of three

different departments in an organisation. The probability of failure for each component is provided in

the table below:

COMPONENT PROBABILITY OF FAILURE

#1 0.12

#2 0.09

#3 0.10

Assume the components operate independently of each other, find the probability that the system

operates.

Solution

Let A be the event that component #1 functions properly.

Let B be the event that component #2 functions properly.

Let C be the event that component #3 functions properly.Since this is a case of independent events, we are looking for:

Question 5

One of the greatest problems in marketing research and other survey fields is the problem of non-

response to surveys. In home interviews the problem arises from absence from home at the time of

visit or, sometimes, refusal to answer questions. A market researcher believes that a respondent willanswer all questions with a probability of 0.94 if found at home. He further believes that the

probability that a given person will be found at home is 0.65. Given this information, what percentage

of the interviews will be successfully completed?

Solution

Let A be the event that the respondent answers all questions.

Let B be the event that the respondent is found at home.

3

#

1

#

3

#

2

0)(09.0)(,12.0)(,90.0)(,91.0)(,88.0)(,

)()()()(

======

+++

CPandBPAPCPBPAPWhere

CBAPCBAPCBAPCBAP

0.97036.

10.091.088.090.009.088.090.091.012.090.091.088.0

)()()()()()()()()()()()(

)()()()(,

=

+++=

+++=

+++

CPBPAPCPBPAPCPBPAPCPBPAP

CBAPCBAPCBAPCBAPNow


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In order for the interview to be successful, we need to have both the events mentioned above take place.

Thus, we are looking for:

Question 6

A firm is supplied with components from three sources A, B and C. It is further known that 50% of the

supplies come from A, 30% come from B and the rest come from C. It is also found from experiencethat, 10% of the components supplied by A are defective, 5% supplied by B are defective and 6% of

those from C are also defective.

If a component is picked at random and is found to be defective, what is the probability that it came

from A?

Solution

Let A be the event that a randomly selected component comes from source A.

Let B be the event that a randomly selected component comes from source B.

Let C be the event that a randomly selected component comes from source C.

Let D be an event that a randomly selected component is defective.

According to the information provided, we have:

Required: To find:

Question 7

ABC Company manufactures items using a maximum of three different operations; shaping, plating

and finishing. A defect can arise in an item during any operation that it undergoes and it may beclassified as minor or major. The following table gives the probabilities of a defect arising during each

of the three operations, which are independent:-

4

%.1.61611.0

65.094.0)(,.65.0)(94.0)/(

:,

)()/()(

or

BAPThusBPandBAP

haveweprovideddatatheFrom

BPBAPBAP

=

=

==

=

.06.0)/(05.0)/(,10.0)/(,2.0)(,3.0)(,5.0)( ====== CDPandBDPADPCPBPAP

0.6494.

2.006.03.005.05.010.0

5.010.0

))/()()/()()/(

))/()/(

=

++

=

++

=

PCCDPBPBDPAPADP

PAADPDAP


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Operation No defect One minor defect One major defect

Shaping 0.7 0.20 0.10

Plating 0.7 0.10 0.20

Finishing 0.65 0.20 0.15

Required:

Calculate the probability that an item which has gone through all three operations contains:-

a) No defects.

b) One minor defect and two major defects.

c) Two defects of unspecified type.

Solution

a) Required probability is given by:

0.7 0.7 0.65

= 0.3185

b) Required probability is given by:

0.20 0.20 0.15 + 0.10 0.10 0.15 + 0.20 0.10 0.20

= 0.0115.

c) Required probability is given by:

0.7 0.3 0.35 + 0.7 0.3 0.35 + 0.65 0.3 0.3

= 0.2055.

Question 8

XS Marketing has found that 85% of the persons selected for its training programme complete the

course. Of those, 60% become productive salespersons, compared with only 10% of those trainees whodo not complete the training programme.

Required:

a) If a new trainee enters the programme, what is the probability that this person will complete theprogramme and also become a productive salesperson?

b) If a salesperson is deemed to be productive, what is the probability that the person has completedthe training programme?

Solution

Let A be the event that a salesperson selected for a training programme completes the coursesuccessfully.

Let B be the event that a salesperson becomes productive.

From the given information, the following probabilities may be recorded:

5

10.0)/(,60.0)/(,15.0)(,85.0)( ==== ABPABPAPAP


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a) Required probability is given by:

b) Using Bayes rule, we can define the required probability as:

Question 9

As items come to the end of a production line, an inspector chooses which items are to go through a

complete inspection. Suppose that in this particular line, 10% of all items produced are defective.Suppose further that, 60% of all defective items go through a complete inspection, and 20% of all

good items go through a complete inspection. Given that an item is completely inspected, what is

the probability that it is defective?

Solution

Let A be the event that an item selected is defective.

Let B be the event that an item selected goes through a complete inspection.

The above information may be written using probability language as follows:

Required probability is given by:

Question 10

b) The figure below is a representation of a system comprised of two subsystems that are said to

operate in parallel. Each subsystem has two components that operate in series, however, they

operate independently of each other. The system will operate properly as long as at least one ofthe subsystems functions properly. The subsystem on the other hand operates when at least one

component works properly. The reliabilities of each components are:

6

.51.0

85.060.0

)()/()(

=

=

= APABPBAP

0.9714.

15.010.085.060.0

85.060.0

)()/()()/(

)()/()/(

=

+

=

+

=

APABPAPABP

APABPBAP

20.0)/(,90.0)(,60.0)/(,10.0)( ==== ABPAPABPAP

0.25

90.020.010.06.0

10.06.0

)()/()()/(

)()/()/(

=

+

=

+

=

APABPAPABP

APABPBAP


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Component Reliability

# 1 0.9

# 2 0.85

# 3 0.95

# 4 0.9

A System Comprised of Two Parallel Subsystem

Subsystem A

Input Output

Subsystem B

Required:Find the probability that the system operates properly.

Solution to part (b)

Let A be the even that component # 1 functions properly

Let B be the even that component # 2 functions properlyLet C be the even that component # 3 functions properly

Let D be the even that component # 4 functions properly

According to the explanation given above, then the whole system will operate properly if at leastone component in any subsystem works properly. Since the components work independently, we are

looking for the following probability.

Note: Here we apply Demorgans rule.

#

1

#

4

#

3

#

2

0.999925.

.10.005.015.010.01)(,

.10.0)(05.0)(,15.0)(,10.0)(,

)()()()(1

)(1)(

=

=

====

=

=

DCBAPTherefore

DPandCPBPAPWhere

DPCPBPAP

DCBAPDCBAP

solutions for exercise 1

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