Financial Risk Manager Part 1
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A life assurance company insures individuals of all ages. A manager compiled the following statistics of the company’s insured persons:
| Age of insured | Mortality (Probability of death) | Portion of company’s insured persons |
|---|---|---|
| 16–20 | 0.04 | 0.10 |
| 21–30 | 0.05 | 0.29 |
| 31–65 | 0.10 | 0.49 |
| 66–99 | 0.14 | 0.12 |
If a randomly selected individual insured by the company dies, calculate the probability that the dead client was age 16–20.
Other
Community
NNikitesh
A
0.172
B
0.04
C
0.168
D
0.145
Explanation:
This is a Bayes' theorem problem where we need to find the conditional probability that a dead client was in the 21-30 age range given that they died.
Step-by-step solution:
Let:
- B₁ = event that client is in age range 16-20
- B₂ = event that client is in age range 21-30
- B₃ = event that client is in age range 31-65
- B₄ = event that client is in age range 66-99
- B = event that client dies
From the table:
- P(B₁) = 0.1, P(B|B₁) = 0.04
- P(B₂) = 0.29, P(B|B₂) = 0.05
- P(B₃) = 0.49, P(B|B₃) = 0.10
- P(B₄) = 0.12, P(B|B₄) = 0.14
Using Bayes' theorem: P(B₂ | B) = [P(B₂) × P(B|B₂)] / [P(B₁) × P(B|B₁) + P(B₂) × P(B|B₂) + P(B₃) × P(B|B₃) + P(B₄) × P(B|B₄)]
Calculation: Numerator: 0.29 × 0.05 = 0.0145
Denominator:
- (0.1 × 0.04) = 0.004
- (0.29 × 0.05) = 0.0145
- (0.49 × 0.10) = 0.049
- (0.12 × 0.14) = 0.0168 Total denominator = 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843
P(B₂ | B) = 0.0145 / 0.0843 = 0.172 ≈ 17.2%
Key insight: This is a classic application of Bayes' theorem for conditional probability, where we're finding the probability of being in a particular age group given that death has occurred. The answer 0.172 corresponds to option A.
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