##### Section 3.3 Example 4f Sheldon Ross on Baye's Theorem.

An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company’s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability .4, whereas this probability decreases to .2 for a person who is not accident prone.

a) If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy?

b) Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone?

a)

We shall obtain the desired probability by first conditioning upon whether or not the policyholder is accident prone. Let A1 denote the event that the policyholder will have an accident within a year of purchasing the policy, and let A denote the event that the policyholder is accident prone. Hence, the desired probability is given by:

$P(A_1)$ = $P(A_1 | A)P(A) + P(A_1 | A^c)P(A^c)$
$= (.4)(.3) + (.2)(.7) = .26$

b) The desired probability is

$P(A | A_1) = \frac{P(AA_1)}{P(A_1)} = \frac{P(A)P(A_1| A)}{P(A_1)} = \frac{(.3)(.4)}{.26} = \frac{6}{13}$