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%.