Chapter 4 Solutions Manual 2021 stA1610 PDF

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For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020Chapter 4: Introduction to ProbabilityTextbook Exercises: An experiment has three steps with three outcomes possible for the first step, two outcomes possible...

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Chapter 4: Introduction to Probability

Textbook Exercises:

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?

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

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

5. Suppose an experiment has five equally likely outcomes: E1, E2, E3, E4, E5. Assign probabilities to each outcome and show that the requirements in equations (4.3) and (4.4) are satisfied. What method did you use?

6. An experiment with three outcomes has been repeated 50 times, and it was learned that E1 occurred 20 times, E2 occurred 13 times, and E3 occurred 17 times. Assign probabilities to the outcomes. What method did you use?

7. A decision-maker subjectively assigned the following probabilities to the four outcomes of an experiment: P(E1) = 0.10, P(E2) = 0.15, P(E3) = 0.40, and P(E4) = 0.20. Are these probability assignments valid? Explain.

8. Let X1 and X2 denote the interest rates (%) that will be paid on one-year certificates of

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deposit that are issued on the first day of next year (year 1) and the following year (year 2) respectively. X1 takes the values 2, 3 and 4 as does X2.

a. How many sample points are there for this experiment? List the sample points. b. Construct a tree diagram for the experiment.

9. A total of 11 Management students, 4 International Management and American Business Studies (IMABS) and 8 International Management and French Studies (IMF) students have volunteered to take part in an Inter-University tournament. a. How many different ways can a team consisting of 8 Management students, 2 IMABS and 5 IMF students be selected? b. If after the team has been selected, 1 Management, 1 IMABS and 2 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? 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.

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?

11. Refer to Exercise 8.

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Given the probabilities tabulated above (where for example 0.3 = P(X1,X2) = P(3,3)), what is the probability of a. X1 < X2? b. X1 = X2? c. X1 ≥ X2?

12. An experiment has four equally likely outcomes: E1, E2, E3, and E4. a. What is the probability that E2 occurs? b. What is the probability that any two of the outcomes occur (e.g. E1 or E3)? c. What is the probability that any three of the outcomes occur (e.g. E1 or E2 or E4)?

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).

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 fi ve 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. For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020 4-3

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f. What method did you use to assign the probabilities requested?

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?

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. Provide the probability of each of the following events.

a. No vacancies. b. At least four vacancies. c. Two or fewer vacancies.

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?

18. Suppose that we have a sample space with fi ve equally likely experimental outcomes: E1, E2, E3, E4, E5. Let

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19. Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E6, E7}, where E1, E2, . . . , E7 denote the sample points. The following probability assignments apply: P(E1) = 0.05, P(E2) = 0.20, P(E3) = 0.20, P(E4) = 0.25, P(E5) = 0.15, P(E6) = 0.10, and P(E7) = 0.05. Let

20. The probability of a mother of one or more children under the age of six being employed is 0.25. The probability she is employed and uses a day care centre is 0.18. The probability that she uses a day care centre is 0.60. What is the probability therefore that: a. she is either employed or uses a day care centre? b. she is neither employed nor uses a day care centre?

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

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. For use with Anderson, Sweeney, Williams, Camm, Cochran, Freeman and Shoesmith, Statistics for Business and Economics 5e, © Cengage EMEA, 2020 4-5

Chapter 4

a. What is P(A ∩ B)? b. What is P(A | 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?

23.

Refer to Exercise 20.

What is the probability that the mother: a. is employed if she uses the day centre? b. uses a day care centre if she is employed? Is use of the day care centre independent of whether the mother is employed or not? Explain. 24. Test marketing involves the duplication of a planned national marketing campaign in a limited geographical area. The test market of a new home sewing machine was conducted among 75 retail outlets, 50 of which were stores affiliated with a major department store chain. In 40 of the retail outlets, the retailer offered customers a free one-year service contract with the purchase of a new sewing machine. Twenty five of the 50 department stores offered customers a free one-year servicing contract. Suppose a customer buys one of the new sewing machines from a participating test-market retailer: a. What is the probability that the customer did not receive a free one year service contract with the purchase of the sewing machine? b. What is the probability that the customer did receive a free one year service contract but did not buy the sewing machine from a department store? c. If it is known that the customer did receive a free one year service contract with the purchase of the sewing machine, what is the probability that the customer purchased the machine from a department store?

25. A sample of convictions and compensation orders issued at a number of Scottish courts was followed up to see whether the offender had paid the compensation to the victim. Details by gender of offender are as follows:

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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?

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)?

27. The prior probabilities for events A1 and A2 are P(A1) = 0.40 and P(A2) = 0.60. It is also known that P(A1 ∩ A2) = 0. Suppose P(B | A1) = 0.20 and P(B | A2) = 0.05. a. Are A1 and A2 mutually exclusive? Explain. b. Compute P(A1 ∩ B) and P(A2 ∩ B). c. Compute P(B). d. Apply Bayes’ theorem to compute P(A1 | B) and P(A2 | B).

28. The prior probabilities for events A1, A2, and A3 are P(A1) = 0.20, P(A2) = 0.50 and P(A3) = 0.30. The conditional probabilities of event B given A1, A2, and A3 are P(B | A1) = 0.50, P(B | A2) = 0.40 and P(B | A3) = 0.30. a. Compute P(B ∩ A1), P(B ∩ A2) and P(B ∩ A3). b. Apply Bayes’ theorem, equation (4.19), to compute the posterior probability P(A2 | B). c. Use the tabular approach to applying Bayes’ theorem to compute P(A1 | B), P(A2 | B) and P(A3 | B).

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Chapter 4

29. 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%. Sometimes the consultant gets it wrong. Your own experience of the consultant is that he is correct 70% of the time when he predicts that the client will default but that 20% of clients who he believes will not default actually do. a. What is the probability that the new client will not default?

30. In 2014, there were 1,775 fatalities recorded on Britain’s roads, 53 of which were for children (Department of Transport, 2015). Correspondingly, serious injuries totalled 22,807 of which 20,778 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? 31. 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?

32. 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.

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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?

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Chapter 4

Chapter 4: Introduction to Probability

Textbook Exercises Solutions: 1.

Number of experimental Outcomes = (3) (2) (4) = 24

2.

6

6

3.

C3 =

P3 =

6! 6 * 5 * 4 = = 20 3! 3 * 2 *1 ABC

ACE

BCD

BEF

ABD

ACF

BCE

CDE

ABE

ADE

BCF

CDF

ABF

ADF

BDE

CEF

ACD

AEF

BDF

DEF

6! = 6 * 5 * 4 = 120 3!

BDF BFD DBF DFB FBD FDB

4.

a. 1st Toss

2nd Toss

3rd Toss H

H T H

(H,H,H)

T (H,H,T) H

(H,T,H)

T (H,T,T) H

T H T

(T,H,H)

T (T,H,T) H

(T,T,H)

T (T,T,T)

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b. Let: H be head and T be tail

(H,H,H)

(T,H,H)

(H,H,T) (T,H,T) (H,T,H) (T,T,H) (H,T,T) (T,T,T)

c. The outcomes are equally likely, so the probability of each outcomes is 1/8.

5.

P(Ei) = 1 / 5 for i = 1, 2, 3, 4, 5 P(Ei)  0 for i = 1, 2, 3, 4, 5 P(E1) + P(E2) + P(E3) + P(E4) + P(E5) = 1 / 5 + 1 / 5 + 1 / 5 + 1 / 5 + 1 / 5 = 1 The classical method was used.

6.

P(E1) = .40, P(E2) = .26, P(E3) = .34 The relative frequency method was used.

7.

No. Requirement (4.4) is not satisfied; the probabilities do not sum to 1. P(E1) + P(E2) + P(E3) + P(E4) = .10 + .15 + .40 + .20 = .85

8.

a. Nine sample points: (X1, X2) = (2,2), (2,3), (2,4), (3,2), (3,3), (4,4), (4,2), (4,3), (4,4) b. Let

p = positive, n = negative, a = approves, and d = disapproves

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Chapter 4

9. a. Initially the number of ways = 11C8*4C2*8C5 = 55440 b. Afterwards, the number of ways becomes 10C8*3C2*6C5 = 810 Required probability = 810/55440 = 0.01461

10. a. Choose a person at random. Have the person taste the four blends and state which is preferred. b. Assign a probability of 1/4 to each blend. We use the classical method of equally likely outcomes here. c. Blend

Probability

1

.20

2

.30

3

.35

4

.15

Total

1.00

The relative frequency method was used.

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11. a. P(X1 P(B). Yes, continue the ad since it increases the probability of a purchase.

b. Estimate the company’s market share at 20%. Continuing the advertisement should increase the market share since P(B | S) = 0.30.

c.

P(B S) =

P(B  S ) .10 = = .333 P(S) .30

The second ad has a bigger effect.

40.

Let

A = lost time accident in current year B = lost time accident previous year

Given: P(B) = 0.06, P(A) = 0.05, P(A | B) = 0.15 a. P(A  B) = P(A | B)P(B) = 0.15(.06) = 0.009 b. P(A  B) = P(A) + P(B) - P(A  B) = 0.06 + 0.05 - 0.009 = 0.101 or 10.1%

41. a. P(Oil) = 0.50 + 0.20 = 0.70 b. Let S = Soil test results

Events

P(Ai)

P(S | Ai)

P(Ai  S)

P(Ai | S)

High Quality (A1)

0.50

0.20

0.10

0.31

Medium Quality

0.20

0.80

0.16

0.50

(A2)

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No Oil (A3)

0.30

.20

1.00

0.06

0.19

P(S) = 0.32

1.00

P(Oil) = 0.81 which is good; however, probabilities now favor medium quality rather than high quality oil.

42. Joint probability distribution H 0.18 0.07 0.25

CC C All

where

.

CC= C= H= S= MW= WW=

credit card cash household sportswear mens wear womens wear

S 0.04 0.12 0.16

MW 0.11 0.07 0.18

WW 0.37 0.04 0.41

All 0.7 0.3 1

Bold figures are given Non-bold have to be derived

a. P(CC  WW) = P(CC) + P(WW) - P(CC ∩ WW) = 0.7 + 0.41 - 0.37 = 0.74 b. P(MW ∩ C) = 0.07 c. P(S|C) =.12/.3 = 0.40 d. P(C|H) = .07/.25 = 0.28

43.

 9  9  9! 9!     5   0  5! 4! 0! 9! = = 0.014706 p (5) =  18!  18   5!13! 5   4  14 4! 14!    0 5 4! 0! 5!13! = 0.23366 p (0) =    = 18!  18    5!13! 5 

44.

Let D = Default

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