SOLUTION: You are given two events, A and B with the following conditions. P(A | B) = 0.30, P(B | A) = 0.60, P(A ∩ B) = 0.18. a) Find P(B) b) Are A and B independent? Why? c) Find

 
P(A∩B) = 0.18 = P(B∩A)

P(A|B) = P(A∩B)/P(B) = 0.18/P(B) = 0.30
             
                            0.18 = (0.30)P(B)

                       0.18/0.30 = P(B)

                             .60 = P(B)

Let's find out.  A and B will be independent  
if and only if P(A∩B) = P(A)∙P(B), so we substitute:
                 0.18 = (0.30)(0.60)
                 0.18 = 0.18

Yes they are equal so A and B are independent.

P(A') = 1-P(A) = 1-0.30 = 0.70

Since B and A are independent so are B and A' (proved below) 

so

P(B∩A’) = P(B)∙P(A’) = (0.60)(0.70) = 0.42.

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For your information here is how we prove that 

if B and A are independent so are B and A'

P(B∩A) = P(B)∙P(A)  by the definition of independence

= P(B)[1-P(A')]   since P(A) = 1-P(A')

= P(B)-P(B)∙P(A')  

So,
 
(1)   P(B)P(A') = P(B)-P(B∩A) 
 
Since B∩A' =  B-B∩A and B∩A ⊂ B, 
 
(2)   P(B∩A') = P(B)-P(B∩A)
 
From (1) and (2), P(B∩A') = P(B)∙P(A'), so B and A' are independent.

Edwin