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 male policyholder is selected at random, what is the chance he is classified as a standard tier?
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A
15%
B
24%
C
30%
D
33%
Explanation:
Explanation
This is a Bayes' Theorem problem where we need to find the probability that a randomly selected male policyholder is from the standard tier.
Given:
- Standard tier: 40% male
- Preferred tier: 50% male
- Ultra preferred tier: 75% male
- Equal number of policyholders in each tier: P(S) = P(P) = P(U) = 1/3
Using Bayes' Theorem:
[P(S|M) = \frac{P(M|S) \times P(S)}{P(M)}]
Where:
- P(M|S) = 0.40 (probability male given standard tier)
- P(S) = 1/3 (probability of being in standard tier)
- P(M) = total probability of being male
Calculate P(M):
[P(M) = P(M|S) \times P(S) + P(M|P) \times P(P) + P(M|U) \times P(U)] [P(M) = 0.40 \times \frac{1}{3} + 0.50 \times \frac{1}{3} + 0.75 \times \frac{1}{3}] [P(M) = \frac{0.40 + 0.50 + 0.75}{3} = \frac{1.65}{3} = 0.55]
Now calculate P(S|M):
[P(S|M) = \frac{0.40 \times \frac{1}{3}}{0.55} = \frac{0.1333}{0.55} = 0.2424 \approx 24.24%]
Therefore, the probability that a randomly selected male policyholder is from the standard tier is 24%.
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