Homework 1 Solution
#1 Due 9/12/05
1. Suppose the sample space S = {1,2,3,4,5,6,7,8,9}, A={1,3,5,7}, B={6,7,8,9},
C={2,4,8} and
D = {1,5,9}.
A. List the elements of the subsets of S that correspond to the following events
(1)
′
A
∩
B
=
{6,8,9}
(2)
(
′
A
∩
B)
∩
C
=
{8}
(3)
′
B
∪
C
=
{1,2,3,4,5,8}
(4)
(
′
B
∪
C)
∩
D
=
{1,5}
5)
′
A
∩
C
=
{2,4,8}
(6)
(
′
A
∩
C)
∩
D
= φ
B. Suppose it is given that
P(A)
=
4
9
,P(B)
=
4
9
,P(C)
=
3
9
,P(D)
=
3
9
,P(A
∩
B)
=
1
9
,P(A
∩
C)
=
0,
P(A
∩
D)
=
1
9
,P(B
∩
C)
=
1
9
,P(B
∩
D)
=
1
9
Intersections of three or four of the subsets A, B, C, D have probability
zero.(Why?)
Find the probability of each of the subsets in part A.
(1)
P(
′
A
∩
B)
=
P(B)

P(A
∩
B)
=
4
9

1
9
=
3
9
(2)
P[(
′
A
∩
B)
∩
C]
=
P(B
∩
C)

P(A
∩
B
∩
C)
=
1
9
(3)
P(
′
B
∪
C)
=
P(
′
B )
+
P(C)

P(
′
B
∩
C)
=
1

P(B)

P(C)
+
P(C)
+
P(B
∩
C)
=
1

P(B)
+
2P(C)
+
P(B
∩
C)
=
1

4
9
+
6
9
+
1
9
=
6
9
(4)