MULTIPLE CHOICE QUESTIONS

Transcription

1 CHAPTER 6 PROBABILITY MULTIPLE CHOICE QUESTIONS In the following multiple choice que s tions, plea s e circle the correc t answ e r. 1. An appro a c h of assigning proba bilities which ass u m e s that all outco m e s of the experi m e n t are equ ally likely is referr e d to as the: a. subjective appro a c h b. objective appro a c h c. classical appro a c h d. relative frequ e n cy appro a c h c 2. If P(A) = 0.84, P(B) = and P(A or B) =0. 9 0, the n P(A and B) is: a b c d c 3. If P(A) = 0.35, P(B) = 0.45 and P(A and B) = , then P(A/B) is: a. 1.4 b. 1.8 c d d 4. If P(A) = 0.20, P(B) = 0.30 and P(A and B) = 0.06, then A and B are: a. dep e n d e n t event s b. indep e n d e n t event s c. mut u ally exclusive event s d. com ple m e n t a r y event s b 5. If A and B are mut u ally exclusive eve nt s with P(A) = 0.70, the n P(B): a. can be any value betw e e n 0 and 1 b. can be any value betw e e n 0 and 0.70 c. cannot be larger than 0.30 d. cannot be det er mi n e d with the inform a tion given c 6. If A and B are indep e n d e n t eve nt s with P(A) = 0.60 and P(A/B) = 0.60, then P(B) is: a

2 72 Chapter Six b c d. Cannot be det er mi n e d with the inform a tion given d 7. If P(A) = 0.65, P(B) = 0.58, and P(A and B) = 0.76, then P(A or B) is: a b c d b 8. If you roll an unbias e d die 50 tim e s, you should expe c t an eve n num b e r to app e a r : a. at least twice in the 50 rolls b. on every other roll c. 25 out of the 50 rolls d. on the aver a g e, 25 out of the 50 rolls d 9. The collection of all possible outco m e s of an experi m e n t is called: a. a simple event b. a sa m pl e spac e c. a sa m pl e d. a population b 10. Which of the following is not an appro a c h to assigning prob a bilities? a. Classical appro a c h b. Trial and error appro a c h c. Relative frequ e n cy appro a c h d. Subjective appro a c h b 11. A useful graphical met h o d of constr uc ting the sa m pl e spac e for an experi m e n t is: a. a tree diagr a m b. a pie chart c. a histogr a m d. an ogive a A sam pl e spac e of an expe ri m e n t consist s of the following outco m e s : 1, 2, 3, 4, 5. Which of the following is a simple eve nt? a. at least 3 b. at mos t 2 c. 3 d. 15 c Suppo s e P(A) = The proba bility of com pl e m e n t of A is:

3 Probability a b c d c 14. Assum e that you invest e d $10,00 0 in each of thre e stocks. Each stock can incre a s e in value, decr e a s e in value, or re m ain the sa m e. Drawing a prob a bility tree for this expe ri m e n t will show that the num b e r of possible outco m e s is: a. 10,00 0 b. 3 c. 9 d. 27 d 15. An experi m e n t consist s of tossing 3 unbia s e d coins simult a n e o u s ly. Drawing a proba bility tree for this experi m e n t will show that the num b e r of simple event s in this experi m e n t is: a. 3 b. 6 c. 9 d. None of the above answ e r s is correct. d 16. If the event s A and B are indep e n d e n t with P(A) = 0.30 and P(B) = 0.40, the n the prob a bility that both eve nt s will occur simult a n e o u s ly is: a b c d b 17. Two event s A and B are said to be inde p e n d e n t if: a. P(A and B) = P(A). P(B) b. P(A and B) = P(A) + P(B) c. P(A/B) = P(B) d. P(B/A) = P(A) a 18. Two event s A and B are said to mut u ally exclusive if: a. P(A/B) = 1 b. P(B/A) = 1 c. P(A and B) = 1 d. P(A and B) = 0 d 19. Which of the following stat e m e n t s is always correct? a. P(A and B) = P(A). P(B) b. P(A or B) = P(A) + P(B)

4 74 Chapter Six c. P(A or B) = P(A) + P(B) + P(A and B) d. P(A) = 1 P ( AC ) d 20. Which of the following is a require m e n t of the proba bilities assign e d to the outco m e s Oi? a. b. c. P ( Oi ) 0 P ( Oi ) 1 0 P ( Oi ) 1 for each i C d. P ( Oi ) = 1 + P (Oi ) c 21. An experi m e n t consist s of thre e stag e s. There are two possible outco m e s in the first stag e, thre e possible outco m e s in the secon d sta g e, and four possible outco m e s in the third sta g e. Drawing a tre e diagr a m for this experi m e n t will show that the total num b e r of outco m e s is: a. 9 b. 24 c. 26 d. 18 b 22. Which of the following stat e m e n t s is correc t given that the eve nt s A and B have nonzer o proba bilities? a. A and B cannot be both indep e n d e n t and mut u ally exclusive b. A and B can be both indep e n d e n t and mut u ally exclusive c. A and B are always indep e n d e n t d. A and B are always mut u ally exclusive a 23. If A and B are mut u ally exclusive eve nt s, with P(A) = 0.20 and P(B) = 0.30, then P(A and B) is: a b c d c 24. If A and B are indep e n d e n t eve nt s with P(A) = 0.60 and P(B) = 0.70, the n the prob a bility that A occurs or B occurs or both occur is: a b. 0.88

5 Probability c d b 25. If A and B are mut u ally exclusive eve nt s with P(A) = 0.30 and P(B) = 0.40, then P(A or B) is: a b c d. None of the above answ e r s is correct c 26. If A and B are indep e n d e n t eve nt s with P(A) = 0.20 and P(B) = , then P(A/B) is: a b c d a 27. If P(A) = 0.25 and P(B) = 0.65, the n P(A and B) is: a b c d. Cannot be det er mi n e d from the inform a tion given d 28. If a coin is toss e d thre e time s and a statisticia n predict s that the proba bility of obt aining thre e head s in a row is 0.125, which of the following ass u m p tio n s is irrelev a n t to his pre diction? a. The event s are dep e n d e n t b. The event s are indep e n d e n t c. The coin is unbias e d d. All of the above ass u m p ti on s are releva n t to his prediction a 29. If an experi m e n t consist s of five outco m e s with P (O1 ) = 0.10, P (O2 ) = 0.20, P (O3 ) = 0.30, P (O4 ) = 0.40, the n P (O5 ) is a b c d d Of the last 500 custo m e r s ent ering a sup er m a r k e t, 50 have purch a s e d a wireles s phon e. If the classical approach for assigning probabilities is used, the probability that the next customer will purchase a wireless phone is a b. 0.90

6 76 Chapter Six c d. None of the above answers is correct c TRUE/FALSE QUESTIONS 31. If the event of inter e s t is A, the proba bility that A will not occur is the com ple m e n t of A. T 32. The prob a bility of event A and eve nt B occurring mus t be equ al to 1. F 33. The relative frequ e n cy appro a c h to prob a bility dep e n d s on the law of large num b e r s. T 34. The annu al estim a t e of the num b e r of de a t h s of infant s is an exa m pl e of the classical appro a c h to proba bility. F 35. The outco m e of a ga m e of roulet t e bas e d on historic al dat a is not an exa m pl e of the relative frequ e n c y appro a c h to proba bility. F 36. You think you have a 90% cha nc e of pas sing your next principle s of accoun tin g exa m. This is an exa m pl e of subjec tive appro a c h to proba bility. T 37. Probability refers to a num b e r betw e e n 0 and 1, which expr e s s e s the cha nc e that an event will occur. T 38. If event A does not occur, then its com pl e m e n t F 39. Marginal proba bility is the proba bility that a given eve nt will occur, with no other event s taken into conside r a tio n. T 40. Condition al proba bility is the prob a bility that an eve nt will occur, given that anot h e r event will also occur. F 41. When we wish to det er mi n e the proba bility that one or mor e of seve r al event s will occur in an expe ri m e n t, we would use addition rules. T AC will also not occur.

7 Probability A physician has five choice s for tre a ting a patie nt ' s infection. After the first choice has bee n mad e, and bec a u s e of inter a c tion betw e e n the prescription drugs used, ther e are only thre e choice s for the final sta g e of tre a t m e n t. Drawing a proba bility tree for this experi m e n t will show that the total num b e r of possibilities for treating this patie nt is 10. F 43. Two or mor e event s are said to be inde p e n d e n t whe n the occurr e n c e of one event has no effect on the proba bility that anot h e r will occur. T 44. Five stud e n t s from a statistics class have form e d a study group. Each may or may not att e n d a study ses sion. Assuming that the me m b e r s will be making indep e n d e n t decisions on whet h e r or not to att e n d, ther e are 32 differe nt possibilities for the com p o sition of the study ses sion. T 45. When event s are mut u ally exclusive, two or mor e of the m can happ e n at the sa m e time. F 46. According to an old song lyric, "love and marria g e go toge t h e r like a hors e and carriag e." Let love be eve nt A and marria g e be eve nt B. Event s A and B are mut u ally exclusive. F 47. When it is not reas o n a bl e to use the classic al appro a c h to assigning prob a bilities to the outco m e s of an expe ri m e n t, and ther e is no history of the outco m e s, we have no altern a tive but to em ploy the subjec tive appro a c h. T 48. Bayes Law allows us to com p u t e condition al proba bilities from other form s of prob a bility. T 49. A useful graphical met h o d experi m e n t is pie chart. F 50. An experi m e n t consist s of tossing 3 unbia s e d coins simult a n e o u s ly. Drawing a prob a bility tree for this expe ri m e n t will show that the num b e r of outco m e s is 9. F 51. Assum e that A and B are indep e n d e n t eve nt s with P(A) = 0.30 and P(B) = The proba bility that both eve nt s will occur simult a n e o u s ly is F 52. Two event s A and B are said to be inde p e n d e n t if P(A and B) = P(A). P(B) T of constr uc ting the sa m pl e spac e for an

8 78 Chapter Six 53. Two event s A and B are said to mut u ally exclusive if P(A and B) = 0. T 54. If event s A and B have nonz e r o prob a bilities, indep e n d e n t and mut u ally exclusive. F 55. If A and B are indep e n d e n t eve nt s with P(A) = 0.35 and P(B) = 0.55, then P(A/B) is F 56. An effective and simpler met ho d of applying the proba bility rules is the prob a bility tree, wher ein the eve nt s in an expe ri m e n t are repr e s e n t e d by lines. T 57. The proba bility of the union of two mut u ally exclusive eve nt s A and B is P(A or B) = 0. F 58. The relative frequ e n cy appro a c h is useful to interpr e t proba bility stat e m e n t s such as thos e hear d from weat h e r forec a s t e r s or scientist s. T 59. Given that event s A and B are indep e n d e n t and that P( A) = 0.9 and P( B/A) = 0.5, then P(A and B) = T 60. Jim and John go to a coffee shop during their lunch bre a k and toss a coin to see who will pay. The prob a bility that John will pay thre e days in a row is T the n they can be both

9 Probability 79 TEST QUESTIONS 61. Abby, Brend a, and Cam e r o n; thre e candid a t e s for the presid e n c y of a college s stud e n t body, are to addr e s s a stud e n t forum. The forum s organizer is to select the order in which the candida t e s will give their spe e c h e s, and mus t do so in such a way that eac h possible order is equ ally likely to be select e d. a. What is the rando m experi m e n t? b. List the outco m e s in the sa m pl e spac e. c. Assign prob a bilities to the outco m e s. d. What is the proba bility that Cam e r o n will spe a k first? e. What is the proba bility that one of the wom e n will spe a k first? f. What is the proba bility that Abby will spe a k before Cam e r o n doe s? a. The rando m experi m e n t is to obs e rv e the order in which the thre e candid a t e s give their spe e c h e s. b. S = {ABC, ACB, BAC, BCA, CAB, CBA}, wher e A=Abby, B=Bre n d a, C=Ca m e r o n. c. The prob a bility assign e d to eac h outco m e is 1/6. d. 1/3 e. 2/3 f. 1/2 62. Suppo s e A and B are two inde p e n d e n t eve nt s for which P(A) = 0.20 and P(B) = a. Find P(A/B ). b. Find P(B/A). c. Find P(A and B). d. Find P(A or B). a b c d A Ph.D. gradu a t e has applied for a job with two univer sitie s: A and B. The gradu a t e feels that she has a 60% cha nc e of rec eiving an offer from university A and a 50% chanc e of receiving an offer from university B. If she receive s an offer from univer sity B, she believe s that she has an 80% chanc e of receiving an offer from university A. a. What is the proba bility that both universitie s will mak e her an offer? b. What is the proba bility that at least one university will mak e her an offer? c. If she receives an offer from university B, what is the proba bility that she will not receive an offer from university A?

10 Chapter Six a. 0.4 b. 0.7 c. 0.2 Ther e are thre e appro a c h e s to det e r mi ning the prob a bility that an outco m e will occur: classical, relative frequ e n c y, and subjective. Which is mos t appro pri at e in det er mi ning the prob a bility of the following outco m e s? a. The une m ploy m e n t rate will rise next mont h. b. Five toss e s of a coin will result in exactly two he a d s. c. An American will win the Frenc h Open Tennis Tourna m e n t in the year d. A rando mly select e d wom a n will suffer a bre a s t canc e r during the coming year. a. subjective b. classical c. subjective d. relative frequ e n cy 65. Suppo s e P(A) = 0.50, P(B) = 0.40, and P(B/A ) = a. Find P(A and B). b. Find P(A or B). c. Find P(A/B ). a b c At the beginning of each year, an inves t m e n t newsl e t t e r pre dict s whet h e r or not the stock mark e t will rise over the coming year. Historical evide nc e reve al s that ther e is a 75% cha nc e that the stock mark e t will rise in any given year. The newslet t e r has pre dict e d a rise for 80% of the years whe n the mark e t actu ally rose, and has pre dict e d a rise for 40% of the years whe n the mark e t fell. Find the proba bility that the newsl e t t e r s prediction for next year will be correct Suppo s e P( AC ) = 0.30, P( B C / A ) = 0.40, and P( B C / AC ) = a. Find P(A and B). b. Find P( B C ) c. Find P(A or B). a b. 0.43

11 Probability 81 c A stand a r d ad mis sion s test was given at thre e locations. One thous a n d stud e n t s took the test at location A, 600 stud e n t s at location B, and 400 stud e n t s at location C. The perc e n t a g e s of stud e n t s from locations A, B, and C, who pas s e d the test wer e 70%, 68%, and 77%, resp e c tiv ely. One stud e n t is select e d at rando m from am on g thos e who took the test. a. What is the proba bility that the selec t e d stud e n t pas s e d the test? b. If the select e d stud e n t pas s e d the test, what is the prob a bility that the stud e n t took the test at location B? c. What is the prob a bility that the selec t e d stud e n t took the test at location C and failed? ANWERS: a b c Sales records of an applianc e store showe d the following dishwa s h e r s sold weekly for each of the last 50 weeks. Numb er of Dishwas h e r s Sold a. b. c. d. e. num b e r of Numb e r of Weeks Define the rando m experi m e n t of inter e s t to the store. List the outco m e s in the sa m pl e spac e Assign prob a bilities to the outco m e s. What appro a c h have you use d in det er mi ning the prob a bilities in part (c)? What is the proba bility of selling at least two dishwa s h e r s in any given week? a. The rando m experi m e n t consist s of obs e rving the num b e r of dishw a s h e r s sold in any given week. b. S = {0, 1, 2, 3, 4} c. Outco m Prob. e

12 82 Chapter Six d. The relative frequ e n cy appro a c h was use d. e. P{2, 3, 4} = 0.30 A wom a n is expec tin g her secon d child. Her doctor has told her that she has a chanc e of having anot h e r girl. If she has anot h e r girl, ther e is a 90% chanc e that she will be taller than the first. If she has a boy, howev e r, ther e is only a 25% chanc e that he will be taller than the first child. Find the prob a bility that the wom a n s secon d child will be taller than the first A survey of a mag a zin e s subscrib e r s indicat e s that 50% own a hom e, 80% own a car, and 90% of the hom e o w n e r s who subs cribe also own a car. What proportion of subs criber s a. own both a car and a hous e? b. own a car or a hous e, or both? c. own neith er a car nor a hous e? a b c Suppo s e A and B are two mut u ally exclusive eve nt s for which P(A) = 0.30 and P(B) = a. Find P(A and B). b. Find P(A or B). c. Are A and B indep e n d e n t eve nt s? Explain using proba bilities. a b c. No, since P(A and B) = P(A ). P(B) = Suppo s e P(A) = 0.10, P(B /A ) = 0.20, and P(B / AC ) = a. Find P(A and B). b. Find P(A and B C ). c. Find P( B C ). d. Find P(A or B). a b c d. 0.46

13 Probability 74. An invest or tells you that in his estim a tion ther e is 75% chanc e particular stock s price over the next thre e weeks. a. Which appro a c h was use d to produc e this figure? b. Interpr e t the 75% proba bility. 83 that a a. The relative frequ e n cy appro a c h. b. We interpr e t the 75% figure to me a n that if we had an infinite num b e r of stocks with exactly the sa m e econo mic and mark e t char a c t e ris tic s as the one the inves t or will buy, 75% of the m will incre a s e in price over the next thre e weeks. 75. The sam pl e spac e of the toss of a fair coin is S = {1, 2, 3, 4, 5, 6}. If the die is balanc e d, each simple eve nt has the sa m e proba bility. Find the prob a bility of the following event s. a. An odd num b e r b. A num b e r less than or equ al to 3 c. A num b e r great e r than or equ al to 5 d. A num b e r betw e e n 2 and 5, inclusive. a. 3/6 b. 3/6 c. 2/6 d. 4/6 76. Suppo s e P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2. a. Find P(A or B). b. Are A and B mut u ally exclusive eve nt s? Explain. c. Are A and B indep e n d e n t eve nt s? Explain. a b. No, since P(A and B) = 0.20 > 0. c. Yes, since P(A and B) = 0.20 = P(A). P(B). 77. Suppo s e P(A) = 0.30, P(B) = 0.50, and P(B /A ) = a. Find P(A and B). b. Find P(A or B). c. Find P(A /B ). a b c. 0.36

14 Chapter Six Is it possible to have two eve nt s for which P(A) = 0.40, P(B) = 0.50, and P(A or B) = 0.30? Explain. No, since P(A or B) mus t be at least as large as P(B). 79. A statistics profes s or classifies his stud e n t s according to their grad e point aver a g e (GPA) and their gend e r. The acco m p a n yi n g table gives the proportion of stud e n t s falling into the various cat e g ori e s. One stud e n t is select e d at rando m. Gend e r Male Fem al e Under GPA Over a. If the stud e n t select e d is fem al e, what is the proba bility that her GPA is betw e e n 2.0 and 3.0? b. If the GPA of the stud e n t selec t e d is over 3.0, what is the proba bility that the stud e n t is male? c. What is the prob a bility that the stud e n t select e d is fem al e or has a GPA und er 2.0 or both? d. Is GPA indep e n d e n t of gend e r? Explain using proba bilities. a b c d. No; since P(male/GPA over 3.0) = P(male) = 0.40 A phar m a c e u t ic al firm has discove r e d a new diagno s tic test for a cert ain dise a s e that has infect e d 1% of the popula tion. The firm has anno u n c e d that 95% of thos e infect e d will show a positive test result, while 98% of thos e not infect e d will show a neg a tive test result. What proportion of test results are correct? Suppo s e P(A) = 0.50, P(B) = 0.30, and P(A or B) = 0.80 a. Find P(A and B). b. Find P(B/A ). c. Are A and B mut u ally exclusive eve nt s : Explain using prob a bilities.

15 Probability 85 a. 0.0 b. 0.0 c. Yes; since P(A and B) = Suppo s e P(A) = 0.40, P(B) = 0.50, and P(A or B) = a. Find P(A and B). b. Find P(B/A ). c. Are A and B indep e n d e n t eve nt s? Explain using proba bilities. a b c. Yes; since P(B/A ) = 0.50 = P(B) 83. An accou n ting firm has rec e n tly recruit e d five gradu a t e s : two me n and thre e wom e n. Two of the grad u a t e s are to be selec t e d at rando m to work in the firm s subur b a n office. a. What is the proba bility that two wom e n will be select e d? b. What is the proba bility that at least one wom a n will be select e d? a b An insur a nc e com p a n y has collect e d the following dat a on the gend e r and marit al stat u s of 300 custo m e r s. Marital Stat us Gend er Male Fem al e Single Married Divorce d Suppo s e that a custo m e r is select e d at rando m. Find the proba bility that the custo m e r select e d is: a. a married fem al e b. not single c. married if the custo m e r is male d. fem al e or divorced e. Are gend e r and marit al stat u s mut u ally exclusive? Explain using prob a bilities. f. Is marit al stat u s indep e n d e n t of gend e r? Explain using proba bilities. a b c

16 86 Chapter Six d e. No, since P(fem al e and marrie d) = > 0. f. No, since P(married / male) = P(marrie d) = QUESTIONS 85 THROUGH 94 ARE BASED ON THE FOLLOWING INFORMATION: An ice crea m vendor sells thre e flavors: chocola t e, straw b e r r y, and vanilla. Forty five perce n t of the sales are chocolat e, while 30% are straw b e r r y, with the rest vanilla flavore d. Sales are by the cone or the cup. The perce n t a g e s of cone s sale s for chocola t e, straw b e r r y, and vanilla, are 75%, 60%, and 40%, resp e c tiv ely. For a rando mly select e d sale, define the following event s : A1 = chocolat e chos e n A2 = strawb e r r y chos e n A3 = vanilla chos e n B B C 85. = ice crea m on a cone = ice crea m in a cup Find the proba bility that the ice crea m was sold on a cone and was a. chocolat e flavor b. straw b e r r y flavor c. vanilla flavor a. P(B and A1 ) = P(B/ A1 ).P( A1 ) = (0.75)(0.4 5) = b. P(B and A2 ) = P(B/ A2 ).P( A2 ) = (0.60)(0.3 0) = 0.18 c. P(B and A3 ) = P(B/ A3 ).P( A3 ) = (0.40)(0. 25) = Find the proba bility that the ice crea m was sold in a cup and was a. chocolat e flavor b. straw b e r r y flavor c. vanilla flavor P( B C and A1 ) = P( B C / A1 ).P( A1 ) = (0.25)(0.4 5) = P( B C and A2 ) = P( B C / A2 ).P( A2 ) = (0.40)(0.3 0) = 0.12 P( B C and A3 ) = P( B C / A3 ).P( A3 ) = (0.60)(0. 25) = Find the proba bility that the ice crea m was sold on a cone.

17 Probability 87 P(B) = P( B and A1 ) + P(B and A2 ) + P(B and A3 ) = = Find the proba bility that the ice crea m was sold in a cup. P( B C ) = 1 P( B) = = Find the proba bility that the ice crea m was chocolat e flavor, given that it was sold on a cone P( A1 /B) = P( A1 and B) / P(B) = / = Find the proba bility that the ice crea m was straw b e r r y flavor, given that it was sold on a cone P( A2 /B) = P( A2 and B) / P(B) = 0.18 / = Find the proba bility that the ice crea m was vanilla flavor, given that it was sold on a cone P( A3 /B) = P( A3 and B) / P(B) = 0.10 / = Find the proba bility that the ice crea m was chocolat e flavor, given that it was sold in a cup P( A1 / B C ) = P( A1 and B C ) / P( B C ) = / = Find the proba bility that the ice crea m was straw b e r r y flavor, given that it was sold in a cup P( A2 / B C ) = P( A2 and B C ) / P( B C ) = 0.12 / = Find the proba bility that the ice crea m was vanilla flavor, given that it was sold in a cup P( A3 / B C ) = P( A3 and B C ) / P( B C ) = 0.15 / = One card is rando mly selec t e d from a deck of 52 playing cards. Let

18 88 Chapter Six A = event card select e d is a nine B = event card select e d is a ten C = event card select e d is a que e n Find P ( AC or B C or C C ) using the addition rule. 1 [(4/52) + (4/52) + (4/52)] = 40/52 = QUESTIONS 96 THROUGH 100 ARE BASED ON THE FOLLOWING INFORMATION: A constr uc tion com p a n y has sub mitt e d bids on two sep a r a t e stat e contr a c t s, A and B. The com p a n y feels that it has a 60% chanc e of winning contr a c t A, and a 50% chanc e of winning contr ac t B. Furthe r m o r e, the com p a n y believe s that it has an 80% chanc e of winning contr act A given that it wins contr a c t B. 96. What is the proba bility that the com p a n y will win both contr a c t s? What is the proba bility that the com p a n y will win at least one of the two contr act s? If the com p a n y wins contr a c t B, what is the proba bility that it will not win contr act A? What is the prob a bility that the com p a n y will win at most one of the two contr act s? What is the proba bility that the com p a n y will win neithe r contr a c t? 0.30 QUESTIONS 101 THROUGH 108 ARE BASED ON THE FOLLOWING INFORMATION: An invest m e n t firm has classified its clients accor ding to their gend e r and the com p o sition of their inves t m e n t portfolio (prim a rily bonds, prim a rily stocks, or a

19 Probability 89 balanc e d mix of bond s and stocks). The propor tions of client s falling into the various cat e g ori es are shown in the following table: Portfolio Compo sition Gend er Male Fem al e Bonds Stocks Balanc e d One client is select e d at rando m, and two eve nt s A and B are define d as follows: A: The client select e d is male. B: The client select e d has a balanc e d portfolio Find the following proba bilities: a. P(A) b. P(B) c. P ( AC ) a b c Expres s each of the following eve nt s in words: a. A or B b. A and B c. A and B C d. AC or B C a. The client b. The client c. The client d. The client 103. select e d select e d select e d select e d either is male or has a balanc e d portfolio or both. is male and has a bala nc e d portfolio. is male and has an unbal a nc e d portfolio. either is fem al e or has an unbal a nc e d portfolio or both. Find the following proba bilities: a. P(A or B) b. P(A and B) c. P(A and B C ) d. P( AC or B C ) a b c d. 0.75

20 Chapter Six Expres s each of the following proba bilities in words: a. P(A/B) b. P(B/A) c. P(A/ B C ) d. P( AC /B) a. The prob a bility that the em ploy e e select e d em ploy e e has a balanc e d portfolio. b. The proba bility that the em ploy e e selec t e d has that the em ploye e is male. c. The prob a bility that the em ploy e e select e d em ploy e e has an unbal a nc e d portfolio. d. The proba bility that the em ploy e e selec t e d em ploy e e has a balanc e d portfolio Find the following proba bilities: a. P(A/B) b. P(B/A) c. P(A/ B C ) d. P( AC /B) a b c d Are A and B indep e n d e n t eve nt s? Explain. No, since P(A/B) = Are A and B C indep e n d e n t eve nt s? Explain. No, since P(A/ B C ) = P(A) = 0.63 P(A) = 0.63 Are A and B C mut u ally exclusive eve nt s? Explain. is male, given that the a bala nc e d portfolio, given is male, given that the is fem al e, given that the

21 Probability 91 No, since P(A and B C ) = 0.38 > 0 QUESTIONS 109 THROUGH 120 ARE BASED ON THE FOLLOWING INFORMATION: A table of joint proba bilities is shown below A1 A2 A3 B B Calculat e the mar gin al prob a bilities of eve nt A P( A1 ) = 0.25, 110. P( A3 ) = 0.35 Calculat e the mar gin al prob a bilities of eve nt B P( B1 ) = 0.60, 111. P( A2 ) = 0.40, P( B2 ) = 0.40 Calculat e P( A1 / B1 ) P( A1 / B1 ) = P( A1 and B1 ) / P( B1 ) = 0.15/ = Calculat e P( A2 / B1 ) P( A2 / B1 ) = P( A2 and B1 ) / P( B1 ) = 0.25 /0.60 = Calculat e P( A3 / B1 ) P( A3 / B1 ) = P( A3 and B1 ) / P( B1 ) = 0.20 / 0.60 = Did your answ er s to Question 111, 112, and 113 sum coinciden c e? Explain. to 1? Is this a

22 92 Chapter Six Yes, they sum to 1. This is not a coincide nc e. The reas o n is that P( A1 and B1 ) + P( A2 and B1 ) + P( A3 and B1 ) = P( B1 ); Ther efor e P( A1 / B1 ) + P( A2 / B1 ) + P( A3 / B1 ) = P( B1 ) / P( B1 ) = Calculat e P( A1 / B2 ) P( A1 / B2 ) = P( A1 and B2 ) / P( B2 ) = 0.10 / 0.40 = Calculat e P( B2 / A1 ) P( B2 / A1 ) = P( B2 and A1 ) / P( A1 ) = 0.10 / 0.25 = Calculat e P( A1 / A2 ) P( A1 / A2 ) = P( A1 and A2 ) / P( A2 ) = 0 / 0.40 = Are the event s A and B inde p e n d e n t? Explain P( A2 / B1 ) = , and P( A2 ) = Since P( A2 / B1 ) P( A2 ), we conclud e that the two event s are dep e n d e n t Calculat e P( A1 or B1 ) P( A1 or B1 ) = P( A1 ) + P( B1 ) P( A1 and B1 ) = = Calculat e P( A1 or B2 ) P( A1 or B2 ) = P( A1 ) + P( B2 ) P( A1 and B2 ) = = 0.55