Introduction to Probability and Statistics for Engineers and Scientists

76 Chapter 3:Elements of Probability

Bayes’ formula may be interpreted as showing us how opinions about these hypotheses
held before the experiment [that is, theP(Fj)] should be modified by the evidence of the
experiment.

EXAMPLE 3.7f A plane is missing and it is presumed that it was equally likely to have
gone down in any of three possible regions. Let 1−αidenote the probability the plane
will be found upon a search of theith region when the plane is, in fact, in that region,
i=1, 2, 3. (The constantsαiare calledoverlook probabilitiesbecause they represent the
probability of overlooking the plane; they are generally attributable to the geographical
and environmental conditions of the regions.) What is the conditional probability that the
plane is in theith region, given that a search of region 1 is unsuccessful,i=1, 2, 3?

SOLUTION LetRi,i=1, 2, 3, be the event that the plane is in regioni; and letEbe the
event that a search of region 1 is unsuccessful. From Bayes’ formula, we obtain

P(R 1 |E)=

P(ER 1 )

P(E)

=

P(E|R 1 )P(R 1 )

∑^3

i= 1

P(E|Ri)P(Ri)

=

(α 1 )(1/3)

(α 1 )(1/3)+(1)(1/3)+(1)(1/3)

=

α 1

α 1 + 2

Forj=2, 3,

P(Rj|E)=

P(E|Rj)P(Rj)

P(E)

=

(1)(1/3)

(α 1 )1/3+1/3+1/3

=

1

α 1 + 2

, j=2, 3

Thus, for instance, ifα 1 =.4, then the conditional probability that the plane is in region
1 given that a search of that region did not uncover it is^16. ■

3.8 INDEPENDENT EVENTS

The previous examples in this chapter show thatP(E|F), the conditional probability of
EgivenF, is not generally equal toP(E), the unconditional probability ofE. In other