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 [arbitrary] | (Probability of death) | Portion of company's insured persons |
|---|---|---|---|
| 16-20 | 0.04 | 0.1 | |
| 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
0.04
B
0.048
C
0.046
D
0.047
Explanation:
This is a Bayes' theorem problem where we need to find the conditional probability that an insured person was in the 16-20 age group given that they died.
Step-by-step calculation:
-
Define events:
- B = Event of death
- B₁ = Event the insured's age is 16-20
- B₂ = Event the insured's age is 21-30
- B₃ = Event the insured's age is 31-65
- B₄ = Event the insured's age is 66-99
-
Given probabilities:
- P(B₁) = 0.1 (portion of company's insured persons aged 16-20)
- P(B₂) = 0.29
- P(B₃) = 0.49
- P(B₄) = 0.12
- P(B|B₁) = 0.04 (mortality rate for age 16-20)
- P(B|B₂) = 0.05
- P(B|B₃) = 0.10
- P(B|B₄) = 0.14
-
Apply Bayes' theorem: P(B₁|B) = [P(B₁) × P(B|B₁)] / [P(B₁) × P(B|B₁) + P(B₂) × P(B|B₂) + P(B₃) × P(B|B₃) + P(B₄) × P(B|B₄)]
-
Calculate numerator: P(B₁) × P(B|B₁) = 0.1 × 0.04 = 0.004
-
Calculate denominator:
- P(B₁) × P(B|B₁) = 0.1 × 0.04 = 0.004
- P(B₂) × P(B|B₂) = 0.29 × 0.05 = 0.0145
- P(B₃) × P(B|B₃) = 0.49 × 0.10 = 0.049
- P(B₄) × P(B|B₄) = 0.12 × 0.14 = 0.0168
- Total = 0.004 + 0.0145 + 0.049 + 0.0168 = 0.0843
-
Final calculation: P(B₁|B) = 0.004 / 0.0843 ≈ 0.04745 ≈ 0.047 (rounded to three decimal places)
Interpretation: Given that a randomly selected insured person has died, there is approximately a 4.7% probability that they were in the 16-20 age group.
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