Find a simple expression for P(A|B) under the following conditions:
(a) A and B are disjoint.
(b) A = B.
(c) A implies B.
(d) B implies A.
Solution:
a) Here , A and B are disjoint.
So, P(A B) = 0
P(A|B) = P(A B) / P(B) = 0/
P(B) = 0
P(A|B) = 0
b) A = B
P(A) = P(B)
P(A|B) = P(A B) /
P(B)
c) A implies B
Here .B is subset of A
So, P(A B) = P(B)
P(A|B) = P(A B) / P(B) = P(B) /
P(B) = 1
d) B implies A.
Here .A is subset of B
P(A B) = P(A)
P(A|B) = P(A B) / P(B) = P(A) /
P(B)
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