Question

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.

Answer 1

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)