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EE 351K Probability, Statistics, and Random Processes
SPRING 2008
Instructor: Shakkottai/Vishwanath
{
shakkott,sriram
}
@ece.utexas.edu
Homework 1
Problem 1
We are given that
P
(
A
) = 0
.
55
,
P
(
B
c
) = 0
.
45
, and
P
(
A
∪
B
) = 0
.
25
. Determine
P
(
B
)
and
P
(
A
∩
B
)
.
Solution :
We have
P
(
B
) = 1

P
(
B
c
) = 1

0
.
45 = 0
.
55
.
Also, by rearranging the formula
P
(
A
∪
B
) =
P
(
A
) +
P
(
B
)

P
(
A
∩
B
)
,
we obtain
P
(
A
∩
B
) =
P
(
A
) +
P
(
B
)

P
(
A
∪
B
) = 0
.
55 + 0
.
55

0
.
25 = 0
.
85
.
Problem 2
Let
A
and
B
be two sets.
(a) Show that
(
A
c
∩
B
c
)
c
=
A
∪
B
and
(
A
c
∪
B
c
)
c
=
A
∩
B
.
(b) Consider rolling a sixsided die once. Let
A
be the set of outcomes where an odd number comes
up. Let
B
be the set of outcomes where a
1
or a
2
comes up. Calculate the sets on both sides of the
equalities in part (a), and verify that the equalities hold.
Solution :
(a) See scanned attachment.
(b) We have
A
=
{
2
,
3
,
5
}
,
B
=
{
3
,
6
}
.
Thus,
A
c
∩
B
c
=
{
2
,
4
,
6
} ∩ {
3
,
4
,
5
,
6
}
=
{
4
,
6
}
,
(
A
c
∩
B
c
)
c
=
{
1
,
2
,
3
,
5
}
,
A
∪
B
=
{
1
,
2
,
3
,
5
}
,
so the first equality is verified. Similarly,
A
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