An insurance company issues life insurance policies in three separate categories: standard, prefferred, and ultra-preferred. Of the company’s policyholders, 50% are standard, 40% are preffered, and 10% are ultra-prefferd. Each standard policyholder has probability 0.010 of dying in the next year, each preferred policyholder has probability 0.005 of dying in the next year, and each ultra-preferred policyholder has probability 0.001 of dying in the next year. Now, suppose a policyholder dies in the next year. What is the probability that the deceased policyholder was ultra-preferred

Answer:0.014 Step-by-step explanation:Hi!Lets call:D = {probability that a policyholder dies next year}std = {policyholder is standard}pref = {policyholder is preferred}ultra = {policyholder is ultra-preferred} We know that:P(std) = 0.5P(pref) = 0.4P(ultra) = 0.1P(D | std) = 0.01P(D | std) = 0.005P(D | std) = 0.001We must find P(ultra | D). We can use Bayes theorem:[tex]P(ultra|D)=P(D|ultra)\frac{P(ultra)}{P(D)}\\P(D) = P(D|std)P(std) + P(D|pref)P(pref)+P(D|ultra)P(ultra) = 0.0071[/tex]Then P(ultra | D) = 0.014