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Byju's Answer
Standard XII
Mathematics
Complement
Let A,B are...
Question
Let
A
,
B
are mutually independent events then show that
¯
¯¯
¯
A
,
B
are also mutually independent events.
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Solution
Given the sets A & B are
mutually independent.
Then
P
(
A
⋂
B
)
=
P
(
A
)
.
P
(
B
)
.
Now, we are to prove
¯
A
& B
are also mutually indendents
Now,
P
(
¯
A
⋂
B
)
=
P
(
B
)
−
P
(
A
⋂
B
)
[
∵
¯
A
⋂
B
=
B
−
A
⋂
B
]
=
P
(
B
)
−
P
(
A
)
.
P
(
B
)
=
P
(
B
)
[
1
−
P
(
A
)
]
=
P
(
B
)
P
(
¯
A
)
[
∵
1
−
P
(
A
)
=
P
(
¯
A
)
]
=
P
(
¯
A
)
.
P
(
B
)
so,
¯
A
,
B
are mutually independent
sets.
Suggest Corrections
0
Similar questions
Q.
Let
A
,
B
,
C
, be three mutually independent events. Consider the two statement
S
1
and
S
2
.
S
1
:
A
and
B
∪
C
are independent.
S
2
:
A
and
B
∩
C
are independent. Then
Q.
Assertion :If the independent events
A
&
B
are such that
0
<
P
(
A
)
<
1
,
0
<
P
(
B
)
<
1
, then
A
&
B
can not be mutually exclusive. Reason: Two events
A
&
B
can not be mutually exclusive, if
P
(
A
∩
B
)
≠
0
.
Q.
If
A
and
B
are two events such that
P
(
A
∪
B
)
′
=
1
6
,
P
(
A
∩
B
)
=
1
4
and
P
(
A
′
)
=
1
4
, then events
A
and
B
are
Q.
Let A and B be two events such that
P
(
¯
¯¯¯¯¯¯¯¯¯¯¯¯
¯
A
∪
B
)
=
1
6
,
P
(
A
∩
B
)
=
1
4
and
P
(
¯
¯¯
¯
A
)
=
1
4
,
w
h
e
r
e
¯
A
stands for the complement of the event A. Then, the events A and B are ?
Q.
If
A
and
B
be two independent events then show that
A
,
B
′
are also independent.
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