Exam P Samples sol2nd

# Exam P Samples sol2nd - SOCIETY OF ACTUARIES/CASUALTY...

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Page 1 of 54 SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE SOLUTIONS Copyright 2005 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study note are taken from past SOA/CAS examinations. P-09-05 PRINTED IN U.S.A. SECOND PRINTING

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Page 2 of 54 1. Solution: D Let event that a viewer watched gymnastics event that a viewer watched baseball event that a viewer watched soccer G B S = = = Then we want to find () () () () ( ) ( ) ( ) ( ) Pr 1 Pr 1P r P r P r P r P r P r P r 1 0.28 0.29 0.19 0.14 0.10 0.12 0.08 1 0.48 0.52 c GBS GBSG BG SB SG B S ⎡⎤ ∪∪ =− ⎣⎦ + + ∩ − ∩ + ∩ ∩ = ++−−−+ = −= -------------------------------------------------------------------------------------------------------- 2. Solution: A Let R = event of referral to a specialist L = event of lab work We want to find P[R L] = P[R] + P[L] – P[R L] = P[R] + P[L] – 1 + P[~(R L)] = P[R] + P[L] – 1 + P[~R ~L] = 0.30 + 0.40 – 1 + 0.35 = 0.05 . -------------------------------------------------------------------------------------------------------- 3. Solution: D First note [ ] [ ] [ ] [ ] [] [ ] [ ] '' ' PA B PA PB PA B PA B PA PB PA B ∪= + + Then add these two equations to get [ ] [ ] [ ] [ ] [ ] ( ) [ ] [ ] ( ) ( ) [] [] '2 ' ' 0.7 0.9 2 1 ' 1.6 2 1 0.6 PA B PA B PA PB PB P A B A B ∪+ = + + ∩+ += + =+ =
Page 3 of 54 4. Solution: A () () [] 12 For 1, 2, let event that a red ball is drawn form urn event that a blue ball is drawn from urn . Then if is the number of blue balls in urn 2, 0.44 Pr[ ] Pr[ ] Pr i i i Ri Bi x R RB B R R B B = = = == + = ∩∪∩ [][] [][] Pr Pr Pr Pr 41 6 6 10 16 10 16 Therefore, 32 3 3 32 2.2 16 16 16 2.2 35.2 3 32 0.8 3.2 4 RR BB x xx x x x + ⎛⎞ =+ ⎜⎟ ++ ⎝⎠ + =+= + += + = = -------------------------------------------------------------------------------------------------------- 5. Solution: D Let N(C) denote the number of policyholders in classification C . Then N(Young Female Single) = N(Young Female) – N(Young Female Married) = N(Young) – N(Young Male) – [N(Young Married) – N(Young Married Male)] = 3000 – 1320 – (1400 – 600) = 880 . -------------------------------------------------------------------------------------------------------- 6. Solution: B Let H = event that a death is due to heart disease F = event that at least one parent suffered from heart disease Then based on the medical records, 210 102 108 937 937 937 312 625 937 937 c c PH F PF ⎡⎤ ∩= = ⎣⎦ and 108 108 625 | 0.173 937 937 625 c c c PHF = =

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Page 4 of 54 7. Solution: D Let event that a policyholder has an auto policy event that a policyholder has a homeowners policy A H = = Then based on the information given, () () ( ) Pr 0.15 Pr Pr Pr 0.65 0.15 0.50 Pr Pr Pr 0.50 0.15 0.35 c c AH A H A H ∩= =−= = and the portion of policyholders that will renew at least one policy is given by ( ) ( ) ( ) ( ) ( ) 0.4 Pr 0.6 Pr 0.8 Pr 0.4 0.5 0.6 0.35 0.8 0.15 0.53 53% cc A H ∩+ + =+ + = = -------------------------------------------------------------------------------------------------------- 100292 01B-9 8. Solution: D Let C = event that patient visits a chiropractor T = event that patient visits a physical therapist We are given that [ ] [ ] Pr Pr 0.14 Pr 0.22 Pr 0.12 CT = = Therefore, [ ] [ ] [ ] [ ] [] [] [] 0.88 1 Pr Pr Pr Pr Pr Pr 0.14 Pr 0.22 2Pr 0.08 C T C T TT T ⎡⎤ =− = = + ⎣⎦ +− ∩∪ or [ ] Pr 0.88 0.08 2 0.48 T =
Page 5 of 54 9. Solution: B Let event that customer insures more than one car event that customer insures a sports car M S = = Then applying DeMorgan’s Law, we may compute the desired probability as follows: () ( ) ( ) () ( ) ()() ( ) ( ) Pr Pr 1 Pr 1 Pr Pr Pr 1 Pr Pr Pr Pr 1 0.70 0.20 0.15 0.70 0.205 c cc M SM S M S M S M S MSS M M ⎡⎤ ∩= = = + ⎣⎦ =− + + = -------------------------------------------------------------------------------------------------------- 10. Solution: C Consider the following events about a randomly selected auto insurance customer: A = customer insures more than one car

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Exam P Samples sol2nd - SOCIETY OF ACTUARIES/CASUALTY...

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