Introduction to Probability and Statistics for Engineers and Scientists

74 Chapter 3:Elements of Probability

characteristic, how certain of the guilt of the suspect should the inspector now be if it turns
out that the suspect is among this group?

SOLUTION LettingGdenote the event that the suspect is guilty andCthe event that he
possesses the characteristic of the criminal, we have

P(G|C)=

P(GC)

P(C)

Now

P(GC)=P(G)P(C|G)

=(.6)(1)

=.6

To compute the probability that the suspect has the characteristic, we condition on whether
or not he is guilty. That is,

P(C)=P(C|G)P(G)+P(C|Gc)P(Gc)

=(1)(.6)+(.2)(.4)

=.68

where we have supposed that the probability of the suspect having the characteristic if
he is, in fact, innocent is equal to .2, the proportion of the population possessing the
characteristic. Hence

P(G|C)=

60

68

=.882

and so the inspector should now be 88 percent certain of the guilt of the suspect. ■

EXAMPLE 3.7e (continued) Let us now suppose that the new evidence is subject to different
possible interpretations, and in fact only shows that it is 90 percent likely that the criminal
possesses this certain characteristic. In this case, how likely would it be that the suspect is
guilty (assuming, as before, that he has this characteristic)?

SOLUTION In this case, the situation is as before with the exception that the probability
of the suspect having the characteristic given that he is guilty is now .9 (rather than 1).
Hence,

P(G|C)=

P(GC)

P(C)

=

P(G)P(C|G)

P(C|G)P(G)+P(C|Gc)P(Gc)