STA222 Exercise 1
1.
I am am try tryin ing g to to dec decid idee whe wheth ther er to mo move ve hous housee or or sta stay y in in my my pre prese sent nt one. one. I mus mustt decide immediately, but unfortunately there is an element of uncertainty in the situation, as my firm is in the middle of reorganization, and may shortly move me to anot anothe herr site site,, clos closer er to my pres presen entt home home than than to the the hous housee I am considering considering buying. I reckon the chances chances of my being being moved at about about 40%. If I am moved, I estimate that annual fares from my present home would cost $200, as opposed to $250 from the new house. To reach the site where I work work at present, on the other hand, currently costs me $300 per year from my present house, but b ut would cost only $200 $ 200 a year from the new ne w house. If there is no price difference between the tow houses, so that I make neither a profit or a loss on the transaction, what would you advise? (move, expected cost $220 as against $260 for staying )
2.
If I do not not put put any any mon money ey in a par parki king ng met meter er whe when n in in park park,, the there re is a oneone-in in-ten chance that a warden warden will notice, notice, and I will be fined fined $25. If I pay now, it will cost me 20 pence. Is it worth trying to get away without paying? (no, expected cost of not paying is $2.50 )
3.
One author suggests that in order to correct our nation’s deteriorating deteriorating rate of productivity, manufacturing firms must engage in effective quality-of-worklife programs for their workers. Suppose that a survey of 74 consumer goods manufacturing firms finds 23 that offer a formal mechanism for worker input to managem management ent decisio decision n processe processess (codeter (codetermina minatio tion), n), 10 have develope developed d flexible work schedules for their employees (flexitime), while 6 engage in both codetermination and flexitime. One firm from among those surveyed is chosen for further analysis of its employee relations programs. (i) (i) What What is is the the prob probab abil ilit ity y the the chos chosen en fir firm m has has both both a cod codet eter ermi mina nati tion on and and a flexitime program?
6 74 (ii) (ii)
What What is the the proba probabi bili lity ty the the chose chosen n firm firm has neit neithe herr a code codete term rmin inat atio ion n nor a flexitime flexitime program?
47 74 (iii) (iii)
Knowi Knowing ng that that the the firm firm has has a flexit flexitim imee progr program am for for its its employ employees ees,, what what is the probability it also engages in codetermination? 3 5
4.
An asse assem mbler bler of of ele elect ctri ricc fans fans uses uses mot motor orss from from two two sour source ces. s. Comp Compan any y A supplies 75% of the motors and company B supplies the other 25% of the motors. Suppose it is is known that 5% of the the motors supplied supplied by company company A are defective, while 3% of the motors supplied by company B are defective. An assembled assembled fan is found to have a defective defective motor. What is the probability probability that company B supplied this motor? (0.17)
5.
I have an old tea set inherited from my great-aunt. At the moment I would get $50 for it if I sold it, which can be invested to give me $65 in a year’s time. If I hang on to the tea set for a year, an antique dealer friend tells me there is a 1 5
chance that this particular type of pottery will become fashionable and I
will be able to sell the tea set for $125. Otherwise the value will remain the same, though my cleaning lady is very clumsy and there is always the chance, if I do decide to keep the tea set for another year, that she will smash it during that year. On past performance I assess the chance of this happening at 10%. What should I do? (Sell now as the expected result of keeping it is only $58.50 ) 6.
The oasis outpost of Abu Ilan, in the heart of the Negev desert, has a population of 20 Beduoin tribesmen and 20 Farima tribesmen. El Kamin, a nearby oasis, has a population of 32 Bedouins and 8 Farima. A lost Israeli soldier, accidentally separated from his army unit, is wandering through the desert and arrives at the edge of one of the oases. The soldier has no idea which oasis he has found, but the first person he spots at a distance is a Bedouin. What is the probability that he wandered into Abu Ilan? (0.3846 )
7.
In major eastern city, 60% of the automobile drivers are 30 years of age or older, and 40% of the drivers are under 30 years of age. Of all drivers 30 years of age or older, 4% will have a traffic violation in a 12-month period. Of all drivers under 30 years of age, 10% will have a traffic violation in a 12month period. Assume that a driver has just been charged with a traffic violation; what is the probability that the driver is under 30 years of age? (0.625)
8.
Developing a small driving range for golfers of all abilities has long been a desire of John Jenkins. John, however, believes that the chance of a successful driving range is only about 40%. A friend of John’s has suggested that he conduct a survey in the community to get a better feeling of the demand for such a facility. There is a 0.9 probability that the research will be favorable if the driving range facility will be successful. Furthermore, it is estimated that there is a 0.8 probability that the marketing research will be unfavorable if indeed the facility will be unsuccessful. John would like to determine the chances of a successful driving range given a favorable result from the marketing survey. (0.75)
9.
A company is in a position to market a new product. A decision on adopting one of the following available options has to be made now. Option 1: Option 2: Option 3:
go ahead and launch the product without market testing. test market first and then launch the product. abandon the launching of the product.
If the product is launched immediately, market research has shown that the probabilities of receiving a strong, weak, and non-existent consumer response will be 40%, 40% and 20% respectively. If a market test is conducted (at a cost of $20000), it is estimated that there will be an equal chance that the test
will indicate a favorable or an unfavorable market condition. If the test indicates a favorable market condition and the product is launched, the probabilities for a strong, weak and non-existent consumer response are estimated to be 72%, 24% and 4% respectively. However, if the test indicates an unfavorable market condition and the product is launched, the probabilities for a strong, weak and non-existent consumer response are estimated to be 8%, 56% and 36% respectively. The payoffs for a strong, weak and non-existent consumer response are, respectively, a profit of $200000, a profit of $50000 and a loss of $150000. All expenses incurred in the research and development of the new product thus far have been ignored in the profit calculations, since they are sunk and therefore can have no bearing on the launching decision. (i) Draw a decision tree for the problem described above, using a square to represent a decision node and a circle to represent a chance (event) node. State the events and the related probabilities at the relevant branches of the tree.
10.
(ii)
Find the expected monetary values (EMV) at each node of the tree. (Option 1: $70000, Option 2: $50000, Option 3: $0)
(iii)
Based on the EMVs calculated, advise the company on the decision to be made. (Choose Option 1)
An electronics company ahs discovered a new sensor for detecting smoke that could prove valuable for the early prevention of fire. At present, it is extremely expensive and the company ahs to decide whether or not to continue with its development. Costs incurred up to this point may be regarded as irrelevant to this decision. It would cost $4 million to develop the sensor fully. The scientists estimate that, at the end of this development phase, the sensor would be ‘very superior’, ‘superior’ or ‘not superior’ to the existing technology with probabilities of 0.3, 0.5 and 0.2 respectively. The company could then market the new product, at a cost of $1 million, or shelve the idea. If the new sensor is marketed there is a risk of increase competition from the other major suppliers. This is estimated to be 80% if it is ‘very superior’, 50% if it is ‘superior’, but only 10% if it is ‘not superior’. The management accountant has estimated the revenue ($ million) from the sales of new sensor to be as follows: Very superior Superior Not superior Increase competition 15 5 1 Same competition 25 10 2 All financial values have been discounted to the present date. (a) Draw the decision tree of the problem described. Using a square for the decision node and a circle for the chance node. (b)
Complete the decision tree with probabilities, costs and revenues, incorporated appropriately.
11.
(c)
Calculate relevant expected values and include these, where appropriate, on the decision tree. (Develop: $4.23, Not develop: $0 )
(d)
Recommend the action the company should take at all the decision points, and explain the basis of your decisions. ( Develop the sensor and market it )
(e)
Find (0.51)
(f)
Find the probability that it is a ‘very superior’ product given that there is increase in competitors. (0.4706 )
the
risk
(probability)
of
increase
competitors.
A departmental store collected data on ‘brand switching’ behavior of their customers over a month. Data are compiled into a transitional matrix shown below. To Brand A Brand B Brand C BrandA From BrandB BrandC
0.800 0.070 0.083
0.100
0.100
0.900
0.030
0.067
0.850
If the market share for Brand A, B and C on 1 st July, 2004 are 0.22, 0.49 and 0.29 respectively, what could be the predicted market share on 1st September, 2004? [ 0.245 0.4766 0.279] 12.
Out of 80 doctors of a hospital, there are a total of 35 surgeons and of these, 20 of them are married. There is only one non-surgeon doctor that is not married. A doctor is randomly selected. (i) What is the probability that the selected doctor is married? (0.8) (ii)
13.
Given that the selected doctor is not married, what is the probability that he/she is a surgeon? (0.9375 )
Monica Britt has enjoyed sailing small boats since she was 7 years old, when her mother started sailing with her. Today, Monica is considering the possibility of starting a company to produce small sailboats for the recreational market. Unlike other mass-produced sailboats, however, these boats will be made specifically for children between the ages of 10 and 15. The boats will be of the highest quality, and extremely stable and the sail size will be reduced to prevent problems of capsizing. Because of the expense involved in developing the initial molds and acquiring the necessary equipment to produce fiberglass sailboats for young children, Monica has decided to conduct a pilot study to make sure that the market for the sailboats will be adequate. She estimates that the pilot study will cost her $10000. Furthermore, the pilot study can be either successful or not successful. Her basic decisions are to build a large manufacturing facility, a small manufacturing facility, or no facility at all. With a favorable market,
Monica can expect to make $90000 from the large facility or $60000 from the smaller facility. If the market is unfavorable, however, Monica estimates that she would lose $30000 with a large facility, while she would lose only $20000 with the small facility. Monica estimates that the probability of a favorable market given a successful pilot study is 0.8. The probability of an unfavorable market given an unsuccessful pilot study result is estimated to be 0.9. Monica feels that there is a 50-50 chance that the pilot study will be successful. Of course, Monica could bypass the pilot study and simply make the decision as to whether to build a large plant, small plant, or no facility at all. Without doing any testing in a pilot study, she estimates that the probability of a successful market is 0.6. What do you recommend? ( Decision is don’t conduct pilot study and build a large manufacturing facility) 14.
George Walls, president of Bradley School, is concerned about declining enrollments. Bradley School is a technical college that specializes in training computer programmers and computer operators. Over the years, there has bee a lot of competition among Bradley School, International Technology, and Career Academy. The three schools compete in providing education in the areas of programming, computer operations, and basic secretarial skills. To gain a better understanding of which of these schools is emerging as a leader, George decided to conduct a survey. His survey looked at the number of students who transferred from one school to the other during their academic careers. On the average, Bradley School was able to retain 65% of those students it originally enrolled. Twenty percent of the students originally enrolled transferred to Career Academy and 15% transferred to International Technology. Career academy had the highest retention rate: 90% of its students remained at Career Academy for their full academic program. George estimated that about half the students who left Career Academy went to Bradley School, while the other half went to International Technology. International Technology was able to retain 80% of its students after they enrolled. Ten percent of the originally enrolled students transferred to Career Academy and the other 10% enrolled in Bradley School. Currently, Bradley School has 40% of the market. Career Academy, a much newer school, has 35% of the market. The remaining market share consists of students attending International Technology. (i) George would like to determine the market share for Bradley School for the next year. [0.3025 0.42 0.2775 ] (ii)
15.
What are the equilibrium market shares for Bradley School, International Technology and Career Academy? [0.1578947 0.578947 0.26316 ]
A study on brand loyalty was conducted among 200 customers of two competitive brands. Each of these people bought exactly one of the brands at any given time. At the start of the study, 140 of them bought Brand A. After a period of 2 months, only 28 of them remained loyal to Brand A. Among those who bought Brand B at the start, 24 remained loyal to Brand B after 2 months.
16.
(i)
With the information given above, form the matrix of transition probabilities. 0.2 0.8 0.6 0.4
(ii)
After 3 periods of transition, the resultant matrix would become 0 .608 x 0.456 0 .544 . The value of x in the resultant matrix is… (0.392)
(iii)
The expected number of customers who remained in Brand A and B after 6 months should be… (87 & 113 )
Company A has a choice between two investments, i.e. either to save a sum of money in a bank and earn an annual interest amounting to $100000 or to invest the same sum of money to develop a new perfume. Because the already existing expertise, development of the new perfume can be regarded as an assured success. However, when the new perfume is launched, there is 70% chance that a similar product of a rival brand will be introduced into the market. In the launching of the new perfume, Company A can adopt one of the three pricing policies. They are: the Expensively Priced (EX) policy with an estimated profit of $600000, the Averagely Priced (AV) policy with an estimated profit of $300000 and the Cheaply Priced (CH) policy with an estimated profit of $200000. At the same time, the rival company can also respond with an equally competitive set of pricing policies, i.e. the Highly Priced (HP) policy, the Medium Priced (MP) policy and Lowly Priced (LP) policy to the new rival brand product. The conditional probability, P(rival pricing policy| Company A’s pricing policy), e.g. P(HP|EX) = 0.6, are complied in the table below.
Company A’s pricing policy
EX AV CH
Rival pricing policy HP MP LP 0.6 0.3 0.1 0.3 0.5 0.2 0 0.4 0.6
The estimated profit, in $’000, resulting from different combinations of pricing policies of Company A and its rival, are as follows.
Company A’s pricing policy
EX AV CH
Rival pricing policy HP MP LP 150 100 90 120 100 60 250 150 100
e.g. when Company A has adopted the pricing policy EX and the rival company has responded with pricing policy HP, the pay-off would be $150000 etc.
(i) (ii)
17.
18.
Draw the decision tree to picture the whole situation. What is the preferred decision for Company A? Why? ( Develop the perfume and use EX policy )
In a markov system, a transition matrix, (A) will change in every period of transition. (B) is a group of independent probabilities. (C) will remain constant in all transitions in the process. (D) is a regular matrix. (E) none of the above.
(C)
A survey on brand preference was conducted among 600 consumers. The following results summarised the movements in their choices after one year.
Brand From A Brand B Brand C (i)
Brand A 120
To Brand B 60
Brand C 20
60
150
90
10
20
70
Write down the matrix of transition probabilities for the study. 0.6 0.2 0.1
(ii)
(iii)
19.
0. 3 0. 5 0.2
0.1
0. 7 0. 3
Now that 20% of the consumers prefer Brand A, 50% prefer Brand B and 30% prefer Brand C . What is the proportion of consumers that prefer Brand B a year later. (0.37 ) Presently, the number of consumers who prefers Brand A and Brand B and Brand C are 150, 350 and 100 respectively. The estimated number of consumers who will buy Brand A, two years from now, will be… (169)
A local fund manager has to decide whether he should invest locally or overseas for the next three years. For investing overseas, due to regulations, he can either invest in the US, Europe or in East Asia. Investments overseas have to encounter foreign exchange risks. There is a 55% chance that the foreign exchange rates would be favourable. Moreover, there is a one-off initialization cost of £10 million payable now. The investment return from any foreign country can be regarded as high or low, with conditional probabilities of 0.6 and 0.4 respectively. The estimated annual real term profits, in £ million, are as follows: US Hig Low h
Europe Hig Low h
East Asia Hig Low h
Foreign exchange favourable Foreign exchange unfavourable
120 105
80 70
140 130
90 80
160 110
70 60
If invested locally, there is a 65% chance of investing in equities and 35% chance in properties. The probability that the local equity sector would give a high return is 0.7 while the probability that the local property sector would give a high return is 0.3. The expected annual real term profit, in £ million, are:
(i) (ii)
(iii) (iv)
STA222.Ex1
Hig Low h Equity 110 60 Property 100 40 Draw a decision tree and calculate all expected profit or loss for the problem. Which is the best decision for the fund manager? Why? ( If foreign exchange rate is unfavorable, invest in Europe, otherwise in East Asia. EMV: 107.7 vs 82.05; 114 vs 100 ) Suppose that the fund manager has decided to invest locally, what is the probability that he will obtain a high profit? (0.56 ) Assuming that the fund manager has invested locally and obtained a high profit, what is the probability that he has invested in equity? (0.8125 )
