A professional organization (for statisticians, of course) sells term life insurance and major medical insurance. Of those who have just life insurance, 70$\%$ will renew next year, and 80$\%$ of of those with only a major medical policy will renew next. However, 90$\%$ of policyholders
who have both types of policy will renew at least one of them next year. Of the policy holders, 75$\%$ have term life insurance, 45$\%$ have major medical, and 20$\%$ have both.
(a) Calculate the percentage of policyholders that will renew at least one policy next year.
(b) If a randomly selected policy holder does in fact renew next year, what is the probability that he or she has both life and major medical insurance?
(a) .765$\\$ (b) .235
all right, we're given some statistics about a company that offers its clients insurance. 70% of those with Onley Life insurance renew their plan for the next year. 80% of those with only medical insurance renew their plans for next year and 90% with both renew their plans for next year. In addition, 75% of these clients have life insurance, 45% of medical insurance and 20% have both. So to start, I'm gonna note probability that a given client on Lee has life insurance, and that's simply the probability that they have life insurance, minus the probability they have both. So that's gonna be 0.75 0.20 point 55 Do the same. With the probability of only having medical insurance, she's gonna be the probability of having medical insurance in general, which is 0.45 minus the probability of having both 0.20 point 25 all right. For the first part, we want to know the probability that a given client renews at least one of their policies for the next year, and we're going to use a lot of total probability for that. So that's gonna be the probability of having only life insurance times the probability that a client renews, given that they have Onley life insurance, Onley There we go. Plus the probability of only having medical insurance times the probability that they were new, given they only have medical insurance, plus the probability that they have both times the probability they renew, given that they have both. All right, if we plug everything in, we get 0.55 times zero 0.7 plus 0.25 time 0.8 waas zero point What's the 0.2? I'm 0.9. If we write this out, that gives us 0.765 All right party. We want to find the probability that somebody that given that they were new, their policy next year the probability they have both kinds of insurance, So ability of both given that they were new for that, we're just going to use based room. So that's gonna be the probability that they have both times the probability that they renew given they have both over probability. But they were new, all right, so the probability they have both 0.2 The probability they renew, given they have both 0.9 On the last part, We computed that the probability that they were new is 0.765 If you compute this out, that is 0.25 Sorry, it's not 0.250 point 235 three and there you have it.