Probability of age of person renewing insurance? Suppose that 5% of policy holders in a certain car insurance company do not renew their policies the following year. From the previous data 90% of people who renew policy are greater than or equal to 40 years old and 60% of people who do not renew the policy are less than 40 years old. a) When you pick one policy holder, what is the probability that the policy holder's age is younger than 40? b) Given that the one you pick is younger than 40, what is the probability that they are not going to renew the policy next year?

Start with writing down the information you are given:

$P(Does Not Renew) \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad= 0.05$
$P(Renew) = 1- P(DoesNotRenew) \quad\quad\quad\;= 0.95$
$P(>= 40\; |\; Renew) \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;= 0.9$
$P(< 40\; |\; Renew) = 1 - P(>=40\;|\;Renew) \quad= 0.1$
$P(< 40\; |\; Does Not Renew) \quad\quad\quad\;\quad\quad\quad\quad\quad= 0.6$

a) By Law Of Total Probability:
$\quad P(<40) = P(<40 \; | \; Renew)P(Renew) + P(<40 \; | \; DoesNotRenew)P(DoesNotRenew)$
$\quad \quad \quad \quad \quad = (0.1)(0.95) + (0.6)(0.05) = 0.125$

b) By Bayes Theorem:
$\quad P(DoesNotRenew\; | < 40)P(<40) = P(<40\;|\;Does Not Renew)P(Does Not Renew)$ $\quad P(DoesNotRenew\; | < 40)(0.125) = (0.6)(0.05)$
$\quad P(DoesNotRenew\; | < 40) = (0.6)(0.05)/0.125$