Correct Answer - A
Since A and B are independent events.
∴
P
(
A
∩
B
)
=
P
(
A
)
P
(
B
)
â‰
0
∴
So, A and B cannot be mutually exclusive events.
Now,
P
(
A
∩
¯¯¯
B
)
=
P
(
A
)
−
P
(
A
∩
B
)
P
⇒
P
(
A
∩
¯¯¯
B
)
=
P
(
A
)
−
P
(
A
)
P
(
B
)
⇒
⇒
P
(
A
∩
¯¯¯
B
)
=
P
(
A
)
{
1
−
P
(
B
)
}
=
P
(
A
)
P
(
¯¯¯
B
)
⇒
and,
P
(
¯¯¯
A
∩
¯¯¯
B
)
=
1
−
P
(
A
∪
B
)
P
⇒
P
(
¯¯¯
A
∩
¯¯¯
B
)
=
1
−
{
1
−
P
(
¯¯¯
A
)
P
(
¯¯¯
B
)
}
[
∵
⇒
A and B are independent]
⇒
P
(
¯¯¯
A
∩
¯¯¯
B
)
=
P
(
¯¯¯
A
)
P
(
¯¯¯
B
)
⇒
So, A and
¯¯¯
B
B
as well as
¯¯¯
A
and
¯¯¯
B
A
are independent events.
Finally,
P
(
A
/
B
)
+
P
(
¯¯¯
A
/
B
)
=
P
(
A
∩
B
)
P
(
B
)
+
P
(
¯¯¯
A
∩
B
)
P
(
B
)
P
⇒
P
(
A
/
B
)
+
P
(
¯¯¯
A
/
B
)
=
P
(
A
∩
B
)
+
P
(
¯¯¯
A
∩
B
)
P
(
B
)
=
P
(
B
)
P
(
B
)
=
1
⇒
Hence, alternative (a) is incorrect.
