- 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?
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
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:
However, P(A) is assumed to equal
and it was shown in (a)
P(A2|AcA1)=.2, we see that