Introduction to Probability and Statistics for Engineers and Scientists

72 Chapter 3:Elements of Probability

However, based on the fact that he has had an accident within a year, we now reevaluate
his probability of being accident prone as follows.

P(A|A 1 )=

P(AA 1 )

P(A 1 )

=

P(A)P(A 1 |A)

P(A 1 )

=

(.3)(.4)

.26

=

6

13

=.4615 ■

EXAMPLE 3.7c In answering a question on a multiple-choice test, a student either knows
the answer or she guesses. Letpbe the probability that she knows the answer and 1−p
the probability that she guesses. Assume that a student who guesses at the answer will be
correct with probability 1/m, wheremis the number of multiple-choice alternatives. What
is the conditional probability that a student knew the answer to a question given that she
answered it correctly?

SOLUTION LetCandKdenote, respectively, the events that the student answers the
question correctly and the event that she actually knows the answer. To compute

P(K|C)=

P(KC)

P(C)

we first note that

P(KC)=P(K)P(C|K)

=p· 1

=p

To compute the probability that the student answers correctly, we condition on whether
or not she knows the answer. That is,

P(C)=P(C|K)P(K)+P(C|Kc)P(Kc)

=p+(1/m)(1−p)

Hence, the desired probability is given by

P(K|C)=

p

p+(1/m)(1−p)

=

mp

1 +(m−1)p

Thus, for example, ifm=5,p=^12 , then the probability that a student knew the answer
to a question she correctly answered is^56. ■