Example

Example
An insurance company that believes that people can be divided inot two distinct classes -- those who are accident prone and those who are not. During any given year an accident-prone person will have an accident with probability .4, whereas the corresponding figure for a non-accident-prone person is .2. (a) If we assume that 30 percent of the population is accident phone, what is the probability that a new policyholder whill have an accident within a year of purchasing a policy?
(b) What is the conditional probability that a new policyholder will have an accident in his or her second year of policy ownership, given that the policyholder has had an accident in the first year?
[Solution:] (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 purchase; and let A denote the event that the policyholder is accident prone. Hence the desired probability, P(A1), is given by
(b)
If we let A be the event that the policyholder is accident prone and we let Ai, i=1,2, be the event that he or she has had an accident in the ith year, then the desired probability P(A2|A1) may be obtained by conditioning on whether or not the policyholder is accident prone, as follows: P(A2|A1)=P(A2|AA1)P(A|A1)+P(A2|AcA1)P(Ac|A1)
Now,
However, P(A) is assumed to equal , and it was shown in (a) that P(A1)=.26. Hence
and thus
Since P(A2|AA1)=.4 and P(A2|AcA1)=.2, we see that