Correct Answer - A
We have,
P
(
A
∩
B
)
=
p
and
P
(
A
)
+
P
(
B
)
+
P
(
B
)
−
2
P
(
A
∩
B
)
=
q
P
⇒
P
(
A
)
+
P
(
B
)
−
2
p
+
q
⇒
⇒
P
(
A
)
+
P
(
B
)
=
2
p
+
q
⇒
⇒
1
−
P
(
¯¯¯
A
)
+
1
−
P
(
¯¯¯
B
)
=
2
p
+
q
⇒
⇒
P
(
¯¯¯
A
)
+
P
(
¯¯¯
B
)
=
2
−
2
p
−
q
⇒
So, alternative (b) is correct.
Now,
{
(
A
∩
B
)
/
(
A
∪
B
)
}
=
P
[
(
A
∩
B
)
∩
(
A
∪
B
)
]
P
(
A
∪
B
)
{
⇒
P
{
(
A
∩
B
)
/
(
A
∪
B
)
}
=
P
(
A
∩
B
)
P
(
A
∪
B
)
⇒
⇒
P
{
(
A
∩
B
)
/
(
A
∪
B
)
}
=
P
(
A
∩
B
)
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
⇒
⇒
P
{
(
A
∩
B
)
/
(
A
∪
B
)
}
=
p
2
p
+
q
−
p
=
p
p
+
q
⇒
So, alternative (c ) is correct.
Finally,
P
(
¯¯¯
A
∩
¯¯¯
B
)
=
P
(
¯¯¯¯¯¯¯¯¯¯
A
∪
B
)
=
1
−
P
(
A
∪
B
)
P
⇒
P
(
¯¯¯
A
∩
¯¯¯
B
)
=
P
(
¯¯¯¯¯¯¯¯¯¯
A
∪
B
)
=
1
−
[
P
(
A
)
+
P
(
B
)
−
P
(
A
∩
B
)
]
⇒
⇒
P
(
¯¯¯
A
∩
¯¯¯
B
)
=
P
(
¯¯¯¯¯¯¯¯¯¯
A
∪
B
)
=
1
−
[
2
p
+
q
−
p
]
=
1
−
p
−
q
⇒
So, alternative (d) is correct.
Hence, alternative (a) is incorrect.