Introduction to Probability and Statistics for Engineers and Scientists

3.7Bayes’ Formula 71

EF

EFcEF

FIGURE 3.6 E=EF∪EFc.

Equation 3.7.1 states that the probability of the eventEis a weighted average of the
conditional probability ofEgiven thatFhas occurred and the conditional probability of
Egiven thatFhas not occurred: Each conditional probability is given as much weight as
the event it is conditioned on has of occurring. It is an extremely useful formula, for its
use often enables us to determine the probability of an event by first “conditioning” on
whether or not some second event has occurred. That is, there are many instances where
it is difficult to compute the probability of an event directly, but it is straightforward to
compute it once we know whether or not some second event has occurred.

EXAMPLE 3.7a An insurance company believes that people can be divided into two
classes — those that are accident prone and those that are not. Their 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 non-accident-prone
person. If we assume that 30 percent of the population is accident prone, what is the prob-
ability that a new policy holder will have an accident within a year of purchasing a policy?

SOLUTION We obtain the desired probability by first conditioning on whether or not the
policy holder is accident prone. LetA 1 denote the event that the policy holder will have
an accident within a year of purchase; and letAdenote the event that the policy holder is
accident prone. Hence, the desired probability,P(A 1 ), is given by

P(A 1 )=P(A 1 |A)P(A)+P(A 1 |Ac)P(Ac)

=(.4)(.3)+(.2)(. 7)=.26 ■

In the next series of examples, we will indicate how to reevaluate an initial probability
assessment in the light of additional (or new) information. That is, we will show how to
incorporate new information with an initial probability assessment to obtain an updated
probability.

EXAMPLE 3.7b Reconsider Example 3.7a and suppose that a new policy holder has an
accident within a year of purchasing his policy. What is the probability that he is accident
prone?

SOLUTION Initially, at the moment when the policy holder purchased his policy, we
assumed there was a 30 percent chance that he was accident prone. That is,P(A)=.3.