60% of an insurer's policyholders are male and 40% are female. The chance of a female having a claim is twice the chance of a male having a claim. Given a randomly selected policyholder has a claim, what's the probability that the policyholder is a male?

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Explanation:

This is a conditional probability problem that can be solved using Bayes' theorem.

Given:

P(Male) = 0.60
P(Female) = 0.40
P(Claim|Female) = 2 × P(Claim|Male)

Let:

P(C|M) = probability of claim given male
P(C|F) = 2 × P(C|M)

Using Bayes' theorem:

P(M|C) = \frac{P(C|M) \times P(M)}{P(C|M) \times P(M) + P(C|F) \times P(F)}

Substitute values:

P(M|C) = \frac{P(C|M) \times 0.60}{P(C|M) \times 0.60 + [2 \times P(C|M)] \times 0.40}

Simplify:

P(M|C) = \frac{0.60 \times P(C|M)}{0.60 \times P(C|M) + 0.80 \times P(C|M)}

P(M|C) = \frac{0.60}{0.60 + 0.80} = \frac{0.60}{1.40} = 0.4286 \approx 43\%

Explanation: The probability that a claimant is male is approximately 43%, which corresponds to option D. This result makes intuitive sense because although males make up 60% of the policyholders, females are twice as likely to file claims, which reduces the conditional probability of a claimant being male.

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