Some Properties of Independent Events

If A and B are Independent events then:

1.  A and Bc are also independent.

2.  Ac and B  are also independent.

3.  Ac and Bc are also independents.

 

I will prove one and you please try to prove the remaining on your own.

let's prove:

3. Ac and Bc are also independents.

if  Ac and Bc are also independents then 

P(A ∩ Bc) = P(Ac).P(Bc)

P(A ∩ Bc) ¬ P(A ∪ B ) = 1 - P(A ∪ B )

                        =  1 - {P(A) + P(B) - P(A ∩ B) }

                        =  1-   {P(A) + P(B) - P(A). P(B) }

                        =  {1- P(A) } -  P(B)  {1- P(A) }

                        = {1 - P(A)}   {1- P(B)}

                        =  P(Ac).P(Bc)

 

Hence Proved.

   

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