Midterm 1 Solution
Stat416/Math416, Spring 2005
J. Li
•
The number of points assigned to each question is indicated before the question. The full
score for the exam is 100.
1. (20) There are two coins. One is a fair coin and the other is a biased coin with probability
of obtaining head
P
(
H
)=0
.
8. A coin is randomly chosen from the two and ﬂipped three
times. What is the probability that three heads are obtained? Suppose that three heads
are obtained, what is the probability that the coin chosen is the biased coin?
Solution:
F
1
: The chosen coin is biased.
F
2
=
F
c
1
: The chosen coin is fair
E
: Three heads are obtained
By total probability formula:
P
(Three heads are obtained)
=
P
(
E
)
=
P
(
F
1
)
P
(
E

F
1
)+
P
(
F
2
)
P
(
E

F
2
)
=
1
2
×
0
.
8
3
+
1
2
×
(
1
2
)
3
=0
.
3185
By Bayes’ Formula
P
(The chosen coin is biased

Three heads are obtained)
=
P
(
F
1

E
)
=
P
(
F
1
)
P
(
E

F
1
)
P
(
F
1
)
P
(
E

F
1
P
(
F
2
)
P
(
E

F
2
)
=(
1
2
×
0
.
8
3
)
/
0
.
3185
.
8038
1
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View Full Document2. Let (
X, Y
) have joint pdf
f
X,Y
(
x, y
)=
c
, for 0
<x<
1, 0
<y<x
2
. The nonzero region
of
f
X,Y
(
x, y
) is also shown by the shaded area in the ﬁgure below.
0
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 Spring '08
 Li
 Probability, Probability theory, Cumulative distribution function, dx, coupons, total probability formula

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