Financial Risk Manager Part 1
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An insurance company writes business in three territories: A, B, and C. They have 150 policyholders in territory A, 250 in territory B, and 300 in territory C. A person is twice as likely to have a claim in territory B than in territory A and 3 times as likely to have a claim in territory C. On average, 50 people have a claim in every policy period. What is the probability that a policyholder in territory C will have a claim during the next policy period?
Other
Community
NNikitesh
A
5%
B
7%
C
10%
D
15%
Explanation:
Explanation
This is a probability problem using the total probability rule. Let's break it down step by step:
Given Information:
- Total policyholders: 150 (A) + 250 (B) + 300 (C) = 700
- Average claims per period: 50
- Let x = P(Claim|A) = probability of claim in territory A
- P(Claim|B) = 2x (twice as likely as A)
- P(Claim|C) = 3x (three times as likely as A)
Using Total Probability Rule: P(Claim) = P(Claim|A) × P(A) + P(Claim|B) × P(B) + P(Claim|C) × P(C)
Where:
- P(A) = 150/700
- P(B) = 250/700
- P(C) = 300/700
- P(Claim) = 50/700 (average claims per period)
Calculation: 50/700 = x × (150/700) + 2x × (250/700) + 3x × (300/700)
Multiply both sides by 700: 50 = 150x + 500x + 900x 50 = 1550x x = 50/1550 = 0.032258
Probability for territory C: P(Claim|C) = 3x = 3 × 0.032258 = 0.096774 ≈ 0.097 = 9.7% ≈ 10%
Verification:
- Expected claims from A: 150 × 0.032258 = 4.84
- Expected claims from B: 250 × 0.064516 = 16.13
- Expected claims from C: 300 × 0.096774 = 29.03
- Total: 4.84 + 16.13 + 29.03 ≈ 50
The probability that a policyholder in territory C will have a claim is approximately 10%, which corresponds to option C.
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