12/10/03 12:48 PM
1
SOLUTION
PEA Final Examination Thursday December 16, 2002
All six questions required for full grade.
Please give explicit numerical answers wherever possible,
on the same page as the question.
Q1
15
Q2
15
Q3
20
Q4
15
Q5
15
Q6
20
TOTAL
100
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12/10/03 12:48 PM
2
1.
A class is taking a multiplechoice test. For any question on the test,
the fraction of examinees that knows the answer to every question is
p
;
1p
is the fraction that guesses. The probability of answering a question
correctly is 1 for an examinee who knows the answer and
1/m
for a
guesser, where
m
is the number of multiplechoice alternatives.
p=0.8, m=5
a. (7) What is the probability that an examinee answers a question
correctly?
P[knows] = p;
P[guesses]=1p;
P[correctknows]=1;
P[correctguesses]=1/m.
P[correct] = P[correctknows]P[knows]
+ P[correctguesses]P[guesses]
= p
×
1 + (1p)
×
1/m = 1/m + p
×
(m1)/m = 0.84
b. (4) Compute the probability that an examinee knew the answer to a
question given that he or she has answered it correctly.
P[knowscorrect] = P[correctknows]P[knows] / P[correct]
= mp / (1 + (m1)p) = 0.95
c. (4) Compute the probability that an examinee answers two specific
questions correctly.
P[correct1,correct2]
=P[correct1,correct2knows]P[knows]
+P[correct1,correct2guesses]P[guesses]
=p + (1p)/m
2.
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 Spring '03
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