If A and B are two events such that P(A) = 3/8 P(B) = 1/2 P(A' n B') =3/8, find the value of P(A' U B')
Answers
C
/B)=2P(A/B
C
)
B
P(B)=P(A
C
/B)
C
15P(A
C
/B
C
)=8P(B/A
C
)
D
P(A/B
C
)=(A∩B)
Step-by-step explanation:
We have
P(A∪B)=P(A)+P(B)−P(A∩B)
⇒
8
5
=
8
3
+
2
1
−P(A∩B)
⇒P(A∩B)=
8
2
=
4
1
Now,
P(A
c
/B)=
P(B)
P(A
c
∩B)
=
P(B)
P(B)−P(A∩B)
=
2
1
2
1
−
4
1
=
4
2
=
2
1
=P(B)
P(A/B
c
)=
P(B
c
)
P(A∩B
c
)
=
1−P(B)
P(A)−P(A∩B)
=
1−
2
1
8
3
−
4
1
=
4
1
=P(A∩B)
Hence,
P(A
c
/B)=2P(A/B
c
)
P(A
c
/B
c
)=
P(B
c
)
P(A
c
∩B
c
)
=
1−P(B)
1−P(A∪B)
=
2
1
1−
8
5
=
4
3
and
P(B/A
c
)=
P(A
c
)
P(A
c
∩B)
=
1−P(A)
P(B)−P(A∩B)
=
1−
8
3
2
1
−
4
1
=
5
2
Thus we see that
P(B/A
c
)
P(A
c
/B
c
)
=
5
2
4
3
=
8
15
⇒8P(A
c
/B
c
)=15P(B/A
c
)