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 territory A and 3 times as likely to have a claim in territory C. On average, 50 people have a claim every policy period. Given a claim occurs, what is the probability it was a policyholder in territory C?
Other
Community
NNikitesh
A
21%
B
33%
C
43%
D
58%
Explanation:
Explanation
This is a conditional probability problem using Bayes' theorem. We need to find P(C|Claim) - the probability that a claim came from territory C given that a claim occurred.
Step 1: Define probabilities
Let:
- x = probability of claim in territory A = P(Claim|A)
- 2x = probability of claim in territory B = P(Claim|B) (twice as likely as A)
- 3x = probability of claim in territory C = P(Claim|C) (three times as likely as A)
Step 2: Calculate prior probabilities
Total policyholders = 150 + 250 + 300 = 700
P(A) = 150/700
P(B) = 250/700
P(C) = 300/700
Step 3: Calculate total probability of claim
Using law of total probability: P(Claim) = P(Claim|A)P(A) + P(Claim|B)P(B) + P(Claim|C)P(C) P(Claim) = x(150/700) + 2x(250/700) + 3x(300/700) P(Claim) = x(150 + 500 + 900)/700 = x(1550)/700
Step 4: Solve for x using average claims
We know on average 50 people have a claim every policy period: P(Claim) = 50/700
So: x(1550)/700 = 50/700 1550x = 50 x = 50/1550 = 0.03226
Step 5: Apply Bayes' theorem
P(C|Claim) = [P(Claim|C) × P(C)] / P(Claim) P(C|Claim) = [3x × (300/700)] / (50/700) P(C|Claim) = [3 × 0.03226 × (300/700)] / (50/700) P(C|Claim) = [0.09678 × 300/700] / (50/700) P(C|Claim) = (29.034/700) / (50/700) P(C|Claim) = 29.034/50 = 0.58068 ≈ 58%
Step 6: Alternative calculation
Since the 700 denominator cancels out: P(C|Claim) = [3x × 300] / 50 = (3 × 0.03226 × 300) / 50 = (0.09678 × 300) / 50 = 29.034/50 = 0.58068
Final answer: 58%
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