Financial Risk Manager Part 1
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An insurance company classifies its policyholders into three tiers – standard, preferred, and ultra preferred. 40% of standard tier policyholders are male, 50% of preferred tier policyholders are male and 75% of ultra preferred tier policyholders are male. There is an equal number of policyholders in each tier. If a policyholder is selected at random, what is the chance she is female?
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Explanation:
Explanation
This is a probability problem using the law of total probability. Let's define the events:
- F: Female policyholder
- S: Standard tier
- P: Preferred tier
- U: Ultra preferred tier
Given information:
- Equal number of policyholders in each tier: P(S) = P(P) = P(U) = 1/3
- Male percentages: 40% in standard, 50% in preferred, 75% in ultra preferred
- Therefore, female percentages: 60% in standard, 50% in preferred, 25% in ultra preferred
Applying the law of total probability:
P(F) = P(F|S) × P(S) + P(F|P) × P(P) + P(F|U) × P(U)
P(F) = 0.60 × (1/3) + 0.50 × (1/3) + 0.25 × (1/3)
P(F) = (0.60 + 0.50 + 0.25) × (1/3)
P(F) = 1.35 × (1/3)
P(F) = 0.45 or 45%
Verification:
- Standard tier: 60% female × 1/3 = 20%
- Preferred tier: 50% female × 1/3 = 16.67%
- Ultra preferred: 25% female × 1/3 = 8.33%
- Total: 20% + 16.67% + 8.33% = 45%
The correct answer is 45%, which corresponds to option D.
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