MATH 166 Spring 2016
1.7
c
Wen
Liu
1.7 Bayes’ Theorem
Bayes’ Theorem: Let E1 , E2 , . . . , En be mutually exclusive events in a sample space S with E1 ∪
E2 ∪ . . . ∪ En = S. If F is any event in S, then for i = 1, 2, . . . , n,
P (Ei ∩ F )
P (Ei |F ) = P
P (Ei ∩ F )
P (Ei )P (F |Ei )
=
P (E1 )P (F |E1 ) + . . . + P (En )P (F |En )
1. A survey involving 800 likely Democratic voters and 300 likely Republican voters asked the
question: Do you support or oppose legislation that would require trigger locks on guns, to prevent misuse by children? The following results were obtained: If a randomly chosen respondent
in the survey answered “support,” what is the probability that he or she is a likely Republican
voter?
2. Box A contains six white marbles and four black marbles. Box B contains three white marbles
and seven black marbles. An experiment consists of first selecting a marble at random from Box
A. The marble is transferred to Box B and then a second marble is drawn from Box B. What
is the probability that the first marble was white given that the second marble was white?
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MATH 166 Spring 2016
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Liu
3. A medical test has been designed to detect the presence of a certain disease. Among people who
have the disease, the probability that the disease will be detected by the test is 0.96. However,
among those who do not have the disease, the probability that the test will detect the presence
of the disease is 0.05. It is estimated that 4% of the population who take this test actually
have the disease. If the test administered to an individual is positive (the disease is detected),
what is the probability that the person actually has the disease?
4. There are three jars that each contain 10 marbles. The first contains 2 white marbles and 8
red marbles, the second 6 white and 4 red, and the third all 10 white. An experiment consists
of first selecting a jar at random. (Assume each jar has an equal probability of being selected.)
After a jar is selected, a marble is randomly drawn from this jar, noting its color. If the marble
drawn was white, find the probability that the third jar was selected.
5. The Office of Admissions and Records of a large western university released the accompanying
information concerning the contemplated majors of its freshman class.
(a) What is the probability that a business student selected at random from the freshman
class is a male?
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MATH 166 Spring 2016
1.7
c
Wen
Liu
(b) What is the probability that a student selected at random from the freshman class is a
female?
(c) What is the probability that a female student selected at random from the freshman class
is majoring in business?
6. Applicants who wish to be admitted to a certain professional school in a large university are
required to take a screening test devised by an educational testing service. From past results,
the testing service has established that 80% of all applicants are eligible for admission and
that 89% of those who are eligible for admission pass the exam, whereas 11% of those who are
ineligible for admission pass the exam.
(a) What is the probability that an applicant for admission passed the exam?
(b) What is the probability that an applicant for admission who passed the exam was actually
ineligible?
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