independent events

Question

if A and B are independent events than P(A / B) = P(A)

will this be true...if yes then how

Answer 1

Firstly lets think about it from logical point of view..

If A and B are independent events , so occurence of A does not depend on occurence of B..So probability of occurence of A will remain unaffected irrespective of occurence of B..

Hence P(A | B) meaning P(A) given P(B) is same as P(A) as A is not dependent on B to occur..

Hence for independent events A and B , P(A | B)  = P(A)..

Secondly let us think from mathematical point of view..

We know ,

P(A | B)     =    P(A ∩ B) / P(B)  

And for independent events  A and B ,  P(A ∩ B)   =  P(A) . P(B)

So ,

 P(A | B)     =    P(A) . P(B) / P(B)  

                  =   P(A)