Answer Keys For The 2008 Edition Of Marcel B. Finan’s “A .
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer Keys for the 2008 Edition ofMarcel B. Finan'sA Probability Course for the ActuariesSecond EditionG. Stolyarov II,ASA, ACAS, MAAA, CPCU, ARe, ARC, API, AIS, AIE, AIAFFirst Edition Published in July-August 2008Second Edition Published in July 2014These answer keys are meant to assist students using Marcel B. Finan's A Probability Coursefor the Actuaries. With Dr. Finan's permission, Mr. Stolyarov wrote solutions for the problemsin his study guide and has endeavored to make the answer keys to each section publiclyavailable. Do the problems at the end of each section and then check your answers with thesekeys.Dr. Finan's study guide is an excellent resource for those preparing to take Actuarial Exam P onprobability.Problems that require numerical answers are answered here, but it is still the responsibility of thestudent to provide his or her own work for these problems. These answers are meant to enablestudents to independently verify the correctness of their reasoning by checking to see if the endresult they obtained is correct. Questions from the study guide that require proofs or diagrams arenot addressed here.The answers in this document are current as of August 2008 and were previously published onAssociated Content / Yahoo! Voices. The contents of A Probability Course for the Actuarieshave been greatly expanded since that time, and some of the problem numbers have changed,while new sections have been added. While updates to these answer keys are possible in thefuture, Mr. Stolyarov’s priority in July 2014 has been to republish the existing answer keyswithin this PDF document, so as to preserve them subsequent to the closure of Yahoo! Voices onJuly 30, 2014.1
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IISection 1.Answer 1.1.A {x:x shows a face with prime number} {2, 3, 5}Answer 1.2a. HHH, HHT, HTH, HTT, TTT, TTH, THT, THHAnswer 1.2b.HTT, TTT, TTH, THTAnswer 1.2c. F in set-builder notation isF {x x is an element of S with more than one head} {x x is an element of S with at leasttwo heads}Answer 1.3. E {HHH, HHT, HTH, HTT, TTH, THT, THH}. The outcomes with more thanone head are, HHH, HHT, HTH. Thus, F {HHH, HHT, HTH,THH}. Thus, F is a subset of E,since F contains only elements that are also in E.Answer 1.4. There are no elements of E and so E ØAnswer 1.9a. 55 tacosAnswer 1.9b. 40 tacosAnswer 1.9c. 10 tacosAnswer 1.10a. 20 studentsAnswer 1.10b. 5 studentsAnswer 1.10c. 11 studentsAnswer 1.10d. 42 studentsAnswer 1.10e. 46 studentsAnswer 1.10f. 46 studentsAnswer 1.11. n(S) 24Section 2.Answer 2.3. (G U B U S)c.Answer 2.5. There are 880 young, single, female policyholders.2
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 2.6. 50% of the population owns an automobile or a house, but not both.Answer 2.7. 5% of visits to a PCP's office result in both lab work and referral to a specialist.Answer 2.8. n(A) 60Answer 2.9. 53% of policyholders will renew at least one policy next year.Answer 2.11. 50 players surveyedAnswer 2.12. 10Answer 2.13a. 3 studentsAnswer 2.13b. 6 studentsAnswer 2.14. Mr. Brown owns 32 chickens.Answer 2.15. 20% of the patients have a regular heartbeat and low blood pressure.Section 3.Answer 3.1a. 100 waysAnswer 3.1b. 900 waysAnswer 3.1c. 5040 waysAnswer 3.1d. 90000 waysAnswer 3.2a. 336 finishing ordersAnswer 3.2b. 6 finishing ordersAnswer 3.3. 6 choices of outfitsAnswer 3.4. 90 waysAnswer 3.6. 36 waysAnswer 3.7. 380 waysAnswer 3.8. 6,497,400 waysAnswer 3.9. 5040 ways3
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 3.10. 3840 waysSection 4.Answer 4.1. m 9 and n 3Answer 4.2a. 456,976 wordsAnswer 4.2b. 358,800 wordsAnswer 4.3a. 15,600,000 possible license platesAnswer 4.3b. 11,232,000 possible license platesAnswer 4.4a. 64,000 possible combinationsAnswer 4.4b. 59,280 possible combinationsAnswer 4.5a. 479,001,600 arrangementsAnswer 4.5b. 604,800 arrangementsAnswer 4.6a. 5Answer 4.6b. 20Answer 4.6c. 60Answer 4.6d. 120Answer 4.7. 20 waysAnswer 4.8a. 362,880 waysAnswer 4.8b. 15,600 waysAnswer 4.9. m 13; n 1 or n 12Answer 4.10. 11,480 possible choicesAnswer 4.11. 300 handshakesAnswer 4.12. 10 waysAnswer 4.13. 28 ways4
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 4.14. 4060 waysAnswer 4.15. The number of permutations of a set of objects is usually greater.Answer 4.16. 125,970 possible juriesAnswer 4.17. 14/2! 43,589,145,600 waysAnswer 4.18. 144 waysSection 5.Answer 5.1. 19,958,400 rearrangementsAnswer 5.2. 9,777,287,520 waysAnswer 5.3. 5.364473777*1028 waysAnswer 5.4. 4,504,501 waysAnswer 5.5. 12,600 possible outcomesAnswer 5.6. 1,260 possible outcomesAnswer 5.7. If all the money needs to be invested, 1771 ways. If not all the money needs to beinvested, 10,626 ways.Answer 5.8. 21 solutionsAnswer 5.9a. 28 distinct termsAnswer 5.9b. 60Answer 5.9c. 729Answer 5.10. 211,876 solutionsAnswer 5.11. About 3.605238748*1016 waysAnswer 5.12a. 136 solutionsAnswer 5.12b. C(15, 2)C(8, 2) 2940 solutionsSection 6.Answer 6.1a. S {1, 2, 3, 4, 5, 6}5
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 6.1b. {2, 4, 6}Answer 6.2. S {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,H), (2,T), (3,1), (3,2), (3,3), (3,4), (3,5),(3,6), (4,H), (4,T), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,H), (6,T)}Answer 6.3. ½ 0.5Answer 6.4a. S {(1, 1), (1,2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4),(4, 1), (4, 2), (4, 3), (4, 4)}Answer 6.4b. Ec {(1, 1), (1,2), (2, 1)}Answer 6.4c. The probability of getting an outcome where the total score is at least six is 3/8; theprobability that the total score is less than 6 is 5/8.Answer 6.4d. ¾ 0.75Answer 6.4e. 5/8 0.625Answer 6.5a. 12/25 0.48Answer 6.5b. P(B) 0Answer 6.5c. P(C) 1Answer 6.5d. 9/25 0.36Answer 6.5e. 1/25 0.04Answer 6.6a. 3/8 0.375Answer 6.6b. ¼ 0.25Answer 6.6c. 1/2 0.5Answer 6.6d.P(11) 0Answer 6.6e. 3/8 0.375Answer 6.6f. 1/8 0.125Answer 6.7a. 4/C(52,13) about 6.29907809*10-12Answer 6.7b. C(4,3)C(48,10)/C(52, 13) about 0.041Answer 6.8. ¼ 0.256
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 6.9. 1/8 0.125Answer 6.10a. 4/n2Answer 6.10b. 5/n2Answer 6.11. 1-200!/[100!*200100]Answer 6.12a. 10 lawyers who are liarsAnswer 6.12b. 2/5 0.4Answer 6.13. 1/10 0.1Section 7.Answer 7.1a. 0.78Answer 7.1b. 0.57Answer 7.1c. 0Answer 7.2. 0.32Answer 7.3. 4/13 about 0.30769Answer 7.4. 5/9 about 0.55555556Answer 7.6a. 2/11 about 0.1818181818Answer 7.6b. 9/11 about 0.81818181818Answer 7.6c. 6/11 about 0.54545454545Answer 7.7. 8/9 about 0.888888888889Answer 7.8. A and B cannot be mutually exclusive.Answer 7.9. 0.52Answer 7.10. 0.05Answer 7.11. P(A) 0.60Answer 7.12. P(T) 0.487
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 7.13. 0.04 1/25Answer 7.14. 0.09 9/100Answer 7.15. 0.5 ½Section 8.Answer 8.1. P(2 red balls) 3/10 0.3P(red ball, then blue ball) 3/10 0.3P(blue ball, then red ball) 3/10 0.3P(2 blue balls) 1/10 0.1Answer 8.2. P(2 red balls) 9/25 0.36P(red ball, then blue ball) 6/25 0.24P(blue ball, then red ball) 6/25 0.24P(2 blue balls) 4/25 0.16Answer 8.3. P(A) 3/5 0.6P(B) 3/10 0.3P(C) 1/10 0.1Answer 8.4. 3/16 0.1875Answer 8.5. 4/9 about 0.44444444Answer 8.6. 1/6 about 0.16666666667Answer 8.7. 3/5 0.6Answer 8.8. 12/25 0.48Answer 8.9. 8/57 about 0.1403508772Answer 8.10. 36/65 about 0.55384615388
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IISection 9.Answer 9.1. 108/625 0.1728Answer 9.2. 0.205Answer 9.3. 7/15 about 0.46666666666667Answer 9.4. ½ 0.5Answer 9.5a. 19/100 0.19Answer 9.5b. 3/5 0.6Answer 9.5c. 31/100 0.31Answer 9.5d. 19/60 about 0.31666667Answer 9.5e. 19/31 about 0.6129032Answer 9.6. 5/33 about 0.15151515151515Answer 9.7. 2/15 about 0.133333333333Answer 9.8. 45 of every 46 parts that make it through the inspection machine are good, but1 of every 46 parts that make it through the inspection machine is defective.Answer 9.9. 72/73 about 0.9863013699Answer 9.10. 7/1912 about 0.0036610879Answer 9.11a. 1/221 about 0.0045248869Answer 9.11b. 1/169 about 0.0059171598Answer 9.12. 1/114 about 0.0087719298Answer 9.13. 80.20202020202%Answer 9.14. 0.476Answer 9.15a. 0.48Answer 9.15b. 0.625Answer 9.15c. 0.19239
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 9.16a. about 0.0821Answer 9.16b. about 0.1268Section 10.Answer 10.1a. 0.26Answer 10.1b. 6/13 or about 0.4615Answer 10.2. 16/101 or about 0.1584Answer 10.3. 1/71 or about 0.01408Answer 10.4. 45/154 or about 0.2922Answer 10.5. 8/19 about 0.42105Answer 10.6. 128/583 about 0.2196Answer 10.7. 190/289 about 0.6574Answer 10.8. 2/5 0.4Answer 10.9. 5/11 or about 0.4545Answer 10.10. 0.66 33/50Answer 10.11a. 0.22 11/50Answer 10.11b. 15/22 about 0.68181818181818Answer 10.12a. 0.56 14/25Answer 10.12b. 4/7 about 0.5714285714Answer 10.13. 5/14 about 0.3571428571Answer 10.14. 1/3Answer 10.15. 15/31 about 0.48387096777Answer 10.16a. 17/140Answer 10.16b. 7/1710
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IISection 11.Answer 11.1a. DependentAnswer 11.1b. IndependentAnswer 11.2. 1/49 about 0.02041Answer 11.3a. 36/169 about 0.21302Answer 11.3b. 48/221 about 0.21719Answer 11.4. 0.72Answer 11.5. 4Answer 11.6. about 0.32872Answer 11.7. 1/10 0.1Answer 11.9. P(C) 0.56; P(D) 0.38Answer 11.10. 0.93Answer 11.11. 0.43Answer 11.13a. The sample space is the set of all possible outcomes. Let N denote the nickel, Ddenote the dime, and Q denote the quarter. Let H denote heads and T denote tails. We willexpress "the dime came up heads" as "DH."S {(NH, DH, QH), (NH, DH, QT), (NH, DT, QH), (NH, DT, QT), (NT, DH, QH),(NT, DH, QT), (NT, DT, QH), (NT, DT, QT)}A {(NH, DH, QH), (NH, DH, QT), (NH, DT, QH), (NT, DH, QH), (NT, DT, QH)}B {(NH, DH, QH), (NH, DT, QH), (NT, DH, QH), (NT, DT, QH)}C {(NH, DH, QH), (NH, DT, QH), (NT, DH, QT), (NT, DT, QT)}Answer 11.13b. P(A) 5/8, P(B) ½, P(C) ½.Answer 11.13c. 0.8 4/5Answer 11.13d. Since P(B and C) P(B)*P(C) ¼, B and C are independent.11
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 11.14. 0.65 13/20Answer 11.15a. 0.70Answer 11.15b. 0.06Answer 11.15c. 0.24Answer 11.15d. 0.72Answer 11.15e. 0.4615Section 12.Answer 12.1. 15:1Answer 12.2. 5/8 0.625Answer 12.3. 1:1Answer 12.4. 4:6Answer 12.5. 1/27 about 0.03704Answer 12.6a. 1:5Answer 12.6b. 1:1Answer 12.6c. 1:0Answer 12.6d. 0:1Answer 12.7. 1:3Answer 12.8a. 3/7 about 0.42857Answer 12.8b. 3/10 0.3Answer 12.9. 2/3Section 13.Answer 13.1a. ContinuousAnswer 13.1b. Discrete12
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 13.1c. DiscreteAnswer 13.1d. ContinuousAnswer 13.2. P(G then B) 15/56; G then B implies X 1.P(G then G) 3/28; G then G implies X 0.P(B then G) 15/56; B then G implies X 1.P(B then B) 5/14; B then B implies X 2.Answer 13.3. 5/36 about 0.1388888889Answer 13.4. 0.85Answer 13.5. (1/2)nAnswer 13.6. P(X J) 2/4 ½Answer 13.7. 2/5 0.4Answer 13.8. 35/36 about 0.97222222222222Answer 13.9a. X(s) f:{HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM} {0, 1, 2,3}Answer 13.9b. P(X 0) 9/100 0.09Answer 13.9c. P(X 1) 9/25 0.36Answer 13.9d. P(X 2) 41/100 0.41Answer 13.9e. P(X 3) 7/50 0.14Answer 13.10. P(X 0) ½; P(X 1) 1/6; P(X 2) 1/12; P(X 3) 1/4Answer 13.11. P(0) 1/(1 e) about 0.2689414214Section 14.Answer 14.1a. The following table shows this probability mass 3/8--3/8---1/813
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 14.2. F(x) 0 for x1/8 for 0 x½ for 1 x7/8 for 2 x1 for x 3Answer 14.3. Formula: P(x) [(6- 7-x )/36]*I 2,3,4,5,6,7,8,9,10,11,12 6Answer 14.4. P(x) (1/6)*I 1,2,3,4,5,6 (x).Answer 14.5. P(x) can be represented via the following table:x:-------0----1----2P(x):--1/4--1/2--1/4P(x) can also be represented as an equation: P(x) [(2- 1-x )/4]*I 0,1,2 (x)Answer 14.6. F(n) 1-(2/3)n 1Answer 14.7a. Pr(X 1) 5/100 1/20 0.05;Pr(X k) k 1Σ ((100-k)(99-k)(98-k)(97-k)/1,806,900,480) for x є integers, 1 x 96xAnswer 14.7b. We can define F(y) as follows:F(y) 0 for y0.5837523669 for 0 y0.9321424769 for 1 y0.9933612861 for 2 y0.9997448142 for 3 y14
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov II0.9999958518 for 4 y1 for y 5We can also express F(y) as the following sum:F(y) Pr(Y y) k 0Σ C(95,10-k)*C(5,k)/C(100,k) for y є integers, 0 y 5yAnswer 14.8. P(x) 3/10 for x -4,2/5 for x 1,3/10 for x 4Answer 14.9. P(x) 4/7 for x -1,3/7 for x 2Answer 14.10a. P(x) 8/27 for x 0,4/9 for x 1,2/9 for x 2,1/27 for x 3Answer 14.10b. F(x) 0 for x8/27 for 0 x20/27 for 1 x20/27 for 1 x26/27 for 2 x1 for x 3Answer 14.11. P(x) 1/36, for x 2,15
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov II1/18, for x 3,1/12, for x 4,1/9, for x 5,5/36, for x 6,1/6, for x 7,5/36 for x 8,1/9 for x 9,1/12 for x 10,1/18 for x 11,1/36 for x 12,0 otherwise.Answer 14.12. p(0) 44/91; p(1) 198/455; p(2) 36/455; p(3) 1/455Section 15.Answer 15.1. 7Answer 15.2. 50/3 about 16.67Answer 15.3. Break even; E(X) 0Answer 15.4. E(X) -1 - 1.00Answer 15.5. E(X) 1/8 about 0.125Since E(X) 0, the owner can expect to make money over an extended period of time usingthis spinner.Answer 15.6. The company should charge each customer a premium of 26 to cover the payoutsand make a 20 profit per person.Answer 15.7. 110Answer 15.8. E(X) -6/11 about - 0.54516
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIThus, we can expect to lose about 54.5 cents per game on average.Answer 15.9. About 896.91Answer 15.10a. 209/783 about 0.2669220945Answer 15.10b. 352/783 about 0.4495530113Answer 15.10c. 16/15 about 1.066666667Answer 15.11a. q 0.3Answer 15.11b. P(X2 2) 0.7Answer 15.11c. 0Answer 15.11d. p(x) 0.2 for x -2,0.3 for x 0,0.1 for x 2.2,0.3 for x 3,0.1 for x 4,0 otherwiseAnswer 15.11e. E(X) 1.12Answer 15.12a. 2400Answer 15.12b. No.Answer 15.13a. p(1) 7/210; p(2) 63/210; p(3) 105/210; p(4) 35/210Answer 15.13b. F(x) 0 for x 1;7/210 for 1 x 2;70/210 for 2 x 3;175/210 for 3 x 4;17
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov II1 for 4 x.Answer 15.13c. E(X) 2.8; Var(X) 0.56Section 16.Answer 16.1a. 1/30 about 0.0333333333Answer 16.1b. 10/3 about 3.33333333333333Answer 16.1c. 127/15 about 8.466666666666667Answer 16.2a. c 1/9Answer 16.2b. p(-1) 2/9, p(1) 1/3, and p(2) 4/9.Answer 16.2c. E(X) 1; E(X2) 7/3 about 2.33333333333333Answer 16.3a. p(x) x/21 for x 1, 2, 3, 4, 5, 6, and p(x) 0 otherwise.Answer 16.3b. 4/7 about 0.5714285714Answer 16.3c. 13/3 about 4.33333333333Answer 16.5. E(F(X)) 0.62Answer 16.6. The expected payment for hospitalization is 220 under this policy.Answer 16.7. 43/1370 about 0.0313868613Answer 16.8. 0.24Answer 16.9a. p(x) 4/7, for x -1,3/7 for x 2,0 otherwise.Answer 16.9b. E(2X) 2Answer 16.10a. P 3C 8A 5S- 300Answer 16.10b. 1,10118
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 16.12. For n 1, E(S) 108. For n 2, E(S) 115.2. For n 3, E(S) 121.6.Section 17.Answer 17.1. 0.45 45%.Answer 17.2. 374.4Answer 17.3a. c 1/55Answer 17.3b. E(X) -12/11 about -1.090909090909;Var(X) 644/605 about 1.06446281Answer 17.4a. p(x) 1/30 for x 1,3/10 for x 2,½ for x 3,1/6 for x 4,0 otherwiseAnswer 17.4b. E(X) 2.8; Var(X) 0.56Answer 17.5a. c 1/30Answer 17.5b. E(X) 10/3; E(X(X-1)) 127/15Answer 17.5c. Var(X) 31/45 about 0.6888889Answer 17.6. E(Y) 21; Var(Y) 144Answer 17.7b. 4/7 about 0.5714285714Answer 17.7c. p(X) 0.24 for x 7,0.16 for x 8,0.18 for x 12,0.42 for x 1519
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 17.7d. E(X) 11.42; Var(X) 12.0036Answer 17.8a. p(x) 4/7, for x -1,3/7 for x 2Answer 17.8b. E(X) 2/7 about 0.2857142857; E(X2) 16/7 about 2.2857142857Answer 17.8c. Var(X) 108/49 about 2.204081633Answer 17.9. Var(X) 9.86; SD(X) about 3.136877428Answer 17.10. E(X) p; Var(X) p(1-p)Section 18.Answer 18.1. 0.38263752Answer 18.2. 27/128 0.2109375Answer 18.3. p(x) 1/8, for x 0,3/8, for x 1,3/8, for x 2,1/8 for x 3,0 otherwiseAnswer 18.4. about 0.0956576009Answer 18.5. 60Answer 18.6. 61/1250 0.0488Answer 18.7. about 0.9245163262Answer 18.8. about 0.1441136931Answer 18.9a. 3/16 0.187520
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 18.9b. 0.5 ½Answer 18.10. about 0.1198380223Answer 18.11. about 0.6241903616Answer 18.12. about 1.82284263*10-5Answer 18.13. SD(X) about 0.6324523697Answer 18.14. E(X2) 154Answer 18.15. 0.135Answer 18.16. 0.784Answer 18.17a. about 0.2321Answer 18.17b. about 0.2649Answer 18.17c. about 0.1238Answer 18.18. about 0.2639Section 19.Answer 19.1. e- 4 about 0.0183156389 probability of no car accident in one day. 8e-8 about0.002683701 probability of 1 car accident in two days.Answer 19.2. About 0.1251100357 probability of receiving 10 calls in 5 minutes.Answer 19.3. e-15 about 3.059023205*10-7 probability of having no errors on a page.Answer 19.4a. about 0.5768099189Answer 19.4b. e-3 about 0.0497870684Answer 19.5a. about 0.9473469827Answer 19.5b. about0.7618966944Answer 19.5c. about 0.160623141Answer 19.6. about 0.761897Answer 19.7a. 1- e-5/6 about 0.565401791521
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 19.7b. about 0.4963317258Answer 19.8. about 0.4231900811Answer 19.9. λ variance 2Answer 19.10. about 7231.30Answer 19.11.about 698.90Answer 19.12. about 0.1550277818Answer 19.13. about 0.758563549Answer 19.14. λ 4Answer 19.15a. about 0.2873Answer 19.15b. about 4.47Answer 19.16. about 0.0838Section 20.Answer 20.1a. 1/10 0.1Answer 20.1b. 9/100 0.09Answer 20.1c. 9k-1/10 kAnswer 20.2. about 0.387420489Answer 20.3. about 0.9158381456Answer 20.4a. 0.001999Answer 20.4b. 1000 timesAnswer 20.5a. 3/64 0.046875Answer 20.5b. 1/256 0.00390625Answer 20.6. about 0.3595464678Answer 20.7a. P(X 3) 4/27 about 0.148148148148; P(X 50) about 7.84164273*10-1022
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 20.7b. E(X) 3Answer 20.8a. P(X n) (0.15)n(0.85) for all whole-number values of n.Answer 20.8b. P(Y n) (0.15)n-1(0.85) for all natural-number values of n.Answer 20.9a. about 0.1198380223Answer 20.9b. about 0.3998629849Answer 20.11. about 0.0527265219Answer 20.12a. 10Answer 20.12b. 0.81Answer 20.13a. X is a geometric distribution with pmf p(x) 0.4(0.6)x-1; x 1, 2, etc.Answer 20.13b. X is a binomial random variable with pmf p(x) C(20, x)(0.60)x(0.40)20-x,where x 0, 1, . , 20.Answer 20.14. 108/625 0.1728Answer 20.15. 96/3125 0.03072Answer 20.16a. C(n-1, 2)(1/13)3(12/13)n-3Answer 20.16b. about 0.017934507Answer 20.17. E(X) 24; Var(X) 120Answer 20.18. 7/64 0.109375Answer 20.19. 3/16 0.1875Answer 20.20. 0.289792Answer 20.21. 0.0221184Answer 20.22a. about 0.1198380223Answer 20.22b. about 0.0253527238Answer 20.23. 2023
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 20.24. about 0.0644972544Answer 20.25. C(t-1, 2)(1/6)3(5/6)t-3Answer 20.26. 1625/4998 about 0.325130052Answer 20.27. 1911/9614 about 0.1987726233Answer 20.28. P(X k) C(6, k)*C(47, 6-k)/C(53, 6), for k 0, 1, 2, 3, 4, 5, or 6Answer 20.29. 80/323 about 0.2476780186Answer 20.30. about 0.0727151097Answer 20.31a. 3/14 0.2142857143Answer 20.31b. mean 3 batteries; Var(X) 3/7 about 0.4285714286Answer 20.32. 6149/7752 about 0.7932146543Answer 20.33. C(2477,3)*C(121373, 97)/C(123850, 100)Answer 20.34. 1/30 about 0.0333333333333333Answer 20.35. about 0.2880Answer 20.36. x 4Answer 20.37. 3/8 0.375Answer 20.38. about 0.956Section 21.Answer 21.1a. p(x) 1/10 for x 2,2/5 for x 6,1/10 for x 10,1/5 for x 11,1/5 for x 15,0 otherwise24
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 21.1b. F(x) 0, for x 2,1/10, for 2 x 6,½, for 6 x 10,3/5, for 10 x 11,4/5, for 11 x 151, for 15 xAnswer 21.1c. E(X) 8.8 centsAnswer 21.2. The cdf of X can be expressed via the following table:Interval for x--(- ,0)---[0,1)-----[1,2)------[2, 3)------[3, 230---1Answer 21.3a. P(X 1) ¼; P(X 2) 1/6; P(X 3) 1/12Answer 21.3b. P(1/2 X 3/2) ½Answer 21.4. p(x) ½ for x 0,1/10 for x 1,1/5 for x 2,1/10 for x 3,1/10 for x 3.5,0 otherwiseAnswer 21.5a. p(x) 0.1, for x -2,0.2, for x 1.1,25
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov II0.3, for x 2,0.4, for x 3,0 otherwiseAnswer 21.5b. P(2 X 3) 0Answer 21.5c. P(X 3) P(X 3) 0.4Answer 21.5d. 4/9 0.444444444444444Answer 21.6a. p 0.1Answer 21.6b. F(X) 0, for x -1.90.1, for -1.9 x -0.10.2, for -0.1 x 20.5, for 2 x 30.6, for 3 x 41, for 4 xAnswer 21.6c. F(0) 0.2; F(2) 0.5; F(F(3.1)) 0.2Answer 21.6d. ½ 0.5Answer 21.6e. E(F(X)) 0.64Answer 21.7a. p(x) 0.3, for x -4,0.4 for x 1,0.3 for x 4,0 otherwiseAnswer 21.7b. Var(X) 9.84; SD(X) about 3.13687742826
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 21.8a. p(x) 1/12 for x 1,1/6 for x 2,1/12 for x 3,1/6 for x 4,1/12 for x 5,1/6 for x 6,1/12 for x 8,1/12 for x 10,1/12 for x 12,0 otherwiseAnswer 21.8b. F(x) 0, for x 11/12, for 1 x 21/4, for 2 x 31/3, for 3 x 41/2, for 4 x 57/12, for 5 x 63/4, for 6 x 85/6, for 8 x 1011/12, for 10 x 121, for 12 xAnswer 21.8c. P(X1/3. P(Xnot the same as F(4).27
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 21.9a. P(X 0) 0; P(X 1) ¼; P(X 2) ¼Answer 21.9b. 7/16 0.4375Answer 21.9c. 3/16 0.1875Answer 21.9d. 3/8 0.375Answer 21.10a. 1/8 0.125Answer 21.10b. about 0.5839466094Answer 21.10c. 1/2 0.5Answer 21.10d. ¼ 0.25Answer 21.11a. e 1/2 P(A) about 0.6065306597Answer 21.11b. e 0.4- e 0.8 P(B) about 0.2209910819Answer 21.11c. (e 0.5- e 0.8)/(e 0.4- e 0.8) P(A B) about 0.7113485948Section 22.Answer 22.1. c 2Answer 22.2a. e 2 about 0.1353352832Answer 22.2b. e 1 - e 2 about 0.2325441579Answer 22.2c. F(x) 1-e x/5if x 0, 0 otherwiseAnswer 22.3. k 3/1000 0.003; 0.027 27/1000Answer 22.4. 15/16 0.9375Answer 22.5a. Yes, F is the cdf of a continuous random variable.Answer 22.5b. f(x) 1/2, 0 x 1,1/6, 1 x 4,0 otherwise28
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 22.6a. k 1Answer 22.6b. 2e-1 about0.7357588823Answer 22.7a. f(x) ex/(ex 1)2Answer 22.7b. e/(e 1) - ½ about 0.2310585786Answer 22.8. About 369.0426555 gallonsAnswer 22.9. 1/9 about 0.111111111111111Answer 22.10. 15/32 0.46875Answer 22.11. 32/243 about 0.1316872428Answer 22.12. 37/64 about 0.578125Answer 22.13. C 0.3Answer 22.14. 511/512 about 0.998046875Answer 22.15. θ 2Answer 22.16.F(x)/F(a) for x a;1 for x a.Section 23.Answer 23.1a. c 1.2Answer 23.1b. F(x) 0, for x -1,0.2x 0.2, for 1 x 0,0.2 0.2x 0.6x2, for 0 x 1,1, for x 1Answer 23.1c. 0.25 ¼29
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 23.1d. E(X) 0.4 2/5Answer 23.2a. a 0.6 3/5 and b 1.2 6/5Answer 23.2b. F(x) 0, for x0.6x 0.4x3, for 0 x 1,1, for x 1Answer 23.3a. E(X) 4Answer 23.3b. E(X) 0Answer 23.3c. E(X) Answer 23.4. E(X) 2/3 about 0.6666666666 Var(X) 2/9 about 0.22222222222Answer 23.5a. E(X) 1/3; SD(X) (2)/3 about 0.4714045208Answer 23.5b. 15/16Answer 23.6. E(lnX) ½Answer 23.7. Var(X) 3/50 0.06Answer 23.8. E(Y) 7/3 about 2.333333333; Var(Y) 34/45 about 0.75555555555556Answer 23.9. 28/15 about 1.8666666667Answer 23.10. About 328.1973799Answer 23.11. E(X) ½ 0.5Answer 23.12. About 123.8291962Answer 23.13. Var(X) 5/36 about 0.138888888889Answer 23.14. E(Y) 3/2 5/2 (πy/4)(cos πy)(1 sin πy)3dyWhen evaluated, E(Y) about 2.2265625.Answer 23.15. 2694.430
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 23.16. (ln 16) aAnswer 23.17. 3ln(2) about 2.079441542Answer 23.18. The mode of X is 2.Answer 23.19. 21/20 1.05Answer 23.20. k 6Answer 23.21. m about 0.8409Answer 23.22a. f X (x) 4x3/106Answer 23.22b. 80Section 24.Answer 24.1a. f(x) 1/4, for 3 x 7, 0 otherwiseAnswer 24.1b. P(XAnswer 24.1c. 1/2Answer 24.2a. f(x) 1/10, 5 x 15, 0 otherwiseAnswer 24.2b. 3/10 0.3Answer 24.2c. E(X) 10; Var(X) 25/3 about 8.333333333333333Answer 24.3a. F(x) 0, for x 0x, for 0 x 11, for x 1Answer 24.4. 1/(n 1)Answer 24.5. ln(2) about 0.6931471806Answer 24.6. 2/3 about 0.6666666666667Answer 24.7. 50031
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 24.8. Var(Y) 128/75 about 1.706666666666667Answer 24.9. SD(X-250) about 403.4357652Answer 24.10a. E(e X) e/2 - 1/2e about 1.175201194Answer 24.10b. Var(e X) about 0.4323323575Answer 24.11. a 3Answer 24.12. 7/10 0.7Answer 24.13. P(X X2) min{1, 1/a}Section 25Answer 25.1a. about 0.2389Answer 25.1b. about 0.1423Answer 25.1c. about 0.6188Answer 25.1d. about 88.12Answer 25.2a. 0.7157Answer 25.2b. about 0.8926Answer 25.2c. about 0.0238Answer 25.3a. 0.5Answer 25.3b. about 0.9876Answer 25.4a. about 0.4772Answer 25.4b. about 0.004Answer 25.5. about 0.0228 57/2500Answer 25.6a. about 0.9452Answer 25.6b. about 0.8185Answer 25.7a. about 0.257832
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 25.7b. about 0.832Answer 25.8. about 0.223584Answer 25.9. The standard deviation of X is 75.Answer 25.10a. P(Y 1060) e-2/[30 (2π)]Answer 25.10b. x 1098Answer 25.11a. P(X 5.5) about 0.4721Answer 25.11b. P(4 X 6.5) about 0.1389Answer 25.11c. P(X 8) about 0.6664Answer 25.11d. P( X 7 4) about 0.5808Answer 25.12. σ about 0.01287Answer 25.13. about 0.8664Answer 25.14a. about 0.1736Answer 25.14b. E(X) about 1070.4Answer 25.15. 0.1151Answer 25.16a. 0.1056Answer 25.16b. 362.84Answer 25.17. 1:18 p.m.Answer 25.18. about 0.1788Answer 25.19a. about 0.4129Answer 25.19b. about 0.2993Answer 25.19c. about 26.81 milliamperesSection 26Answer 26.1. 1-e-0.9 about 0.593430340333
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 26.3. 1- e 0.5 about 0.3934693403Answer 26.4. 1-e-1/8 about 0.1175030974Answer 26.5. e-3/5 about 0.5488116361Answer 26.6a. e-5/3 about 0.1888756028Answer 26.6b. e-2/3- e-4/3 about 0.2498199809Answer 26.7a. e-2 about 0.1353352832Answer 26.7b. e-4/3- e-7/3 about 0.1666251703Answer 26.7c. e-3 about 0.0497870684Answer 26.8. about 0.1339745962Answer 26.9a. e-1.68 about 0.186373976Answer 26.9b. e-0.015- e-0.03 about 0.0146664061Answer 26.10. about 0.4348589372Answer 26.11. about 10255.89899Answer 26.12. E(X) 2 3e-2/3E(X) about 3.540251357Answer 26.13. x about 5644.226968Answer 26.14. m about 173.2867951Answer 26.15. about 0.4204482076Answer 26.16. about 8.325475924Answer 26.17. about 0.0181327887Section 27Answer 27.1. about 0.1238883826Answer 27.2. E(X) 3/2 1.5; Var(X) ¾ 0.7534
Answer Keys for the 2008 Edition of Marcel B. Finan’s “A Probability Course for the Actuaries”Second Edition – G. Stolyarov IIAnswer 27.3. about 0.0947578682Answer 27.4. about 0.4191864763Answer 27.5. about 0.014387678Answer 27.6a. f(x) (1/16)x2e-0.5x, x 0, 0 otherwise;E(X) 6 million kilowatt hours.Answer 27.6b. P(X 12) 12 (1/16)x2e-0.5xdxAnswer 27.7. 480Answer 27.8a. k 60Answer 27.8b. E(X) 4/7 about 0.5714285714; Var(X) 3/98 about 0.0306122449Answer 27.9a. F(X) 0 for x3x2-2x3 for 0 x 1,1 for x 1Answer 27.9b. 99/250 0.396Answer 27.10a. f(x) 60x3(1-x)2 for 0 x 1, 0 otherwise.Answer 27.10b. F(X) 0 for x 0,60x3 120x4 60x5, for 0 x 1,1 for x 1.Answer 27.11. 0.47178Answer 27.12a. f(x) 43.6486021x1.0952(1-x)4.6648, for 0 xAnswer 27.12b. about 0.0829070132Answer 27.13a. k 12Answer 27.13b. 0.6875 11/1635
Answer K
Dr. Finan's study guide is an excellent resource for those preparing to take Actuarial Exam P on probability. . within this PDF document, so as to preserve them subsequent to the closure of Yahoo! Voices on July 30, 2014. . P(T) 0.48. Answer Keys for the 2008 Edition of Marcel
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