8.8 (Notes)
In a quality control inspection items are classi
f
ed as having a minor defect, a major
defect, or as being acceptable. A carton of 10 items contains 2 with minor defects,
1 with a major defect, and 7 acceptable. Three items are chosen at random without
replacement. Let
X
be the number selected with minor defects and
Y
be the number
with major defects.
a)
Find the joint probability function of
X
and
Y
.
b)
Find the marginal probability functions of
X
and of
Y
.
c)
Evaluate numerically
P
(
X
=
Y
)
and
P
(
X
=1

Y
=0)
.
Denote the possible outcomes
M
(major),
m
(minor),
A
(acceptable). We take a sample of
size 3 from the set
{
M, m, m, A, A, A, A, A, A, A
}
.
Then
f
(
x,
0) =
P
(
X
=
x, Y
=0)=
¡
2
x
¢¡
7
3
−
x
¢
¡
10
3
¢
,x
=0
,
1
,
2
f
(
x,
1) =
P
(
X
=
x, Y
=1)=
¡
2
x
¢¡
7
2
−
x
¢
¡
10
3
¢
,
1
,
2
(b)
P
(
X
=
x
)=
¡
2
x
¢¡
7
3
−
x
¢
+
¡
2
x
¢¡
7
2
−
x
¢
¡
10
3
¢
,
1
,
2
(c)
P
(
X
=
Y
f
(0
,
0) +
f
(1
,
1) =
¡
2
0
¢¡
7
3
¢
¡
10
3
¢
+
¡
2
1
¢¡
7
2
−
1
¢
¡
10
3
¢
=
¡
7
3
¢
+14
¡
10
3
¢
=
49
120
.
40833
P
(
Y
¡
9
3
¢
¡
10
3
¢
=
7
10
P
(
X

Y
f
(1
,
0)
P
(
Y
=
¡
2
1
¢¡
7
3
−
1
¢
¡
10
3
¢
7
10
=
1
2
2