THEOREMS ON PROBABILITY OF EVENTS( PART -2)

[If you are beginner in probability then you should read Probability Theorem (Part-1) before Read the below content]

THEOREM- 4:

                   For any 3 events A , B , C  P( A ⋃ B | C ) = P( A | C ) + P( B | C ) P( A ⋂ B | C )

 

PROOF:

      We have P( A ሀ B ) = P( A ) + P( B ) - P( A ⋂ B )

 ⇒P[( A ⋂ C) ⋃ ( B ⋂ C )] = P( A ⋂ C ) + P( B ⋂ C )     - P( A ⋂ B ⋂ C )

         Dividing both sides by P( C ), we get

 ⇒ P[( A ⋂ C) ⋃ ( B ⋂ C )] / P( C ) = P( A ⋂ C ) / P(         C ) + P( B ⋂ C ) / P( C )  - P( A ⋂ B ⋂ C ) / P( C )                                      , P( C ) > 0

                       

 ⇒  P[( A  ⋃ B ) ⋂ C ] / P( C ) = P( A | C ) + P( B | C ) -  P( A ⋂ B | C )

 ⇒   P[( A  ⋃ B ) | C ] = P( A | C ) + P( B | C ) - P( A ⋂ B | C ) 

                                                                        [Proved]

THEOREM - 5:

           For any 3 events A , B , C  P(A ⋂ B' | C ) + P( A ⋂ B | C ) = P( A | C ) , B' = complementary of event B.

PROOF :

          Given,

         P(A ⋂ B' | C ) + P( A ⋂ B | C )

     = P(A ⋂ B' ⋂ C ) / P( C ) + P( A ⋂ B ⋂ C ) / P( C )

     = [ P(A ⋂ B' ⋂ C ) + P( A ⋂ B ⋂ C ) ] / P( C ) 

     =  P(A ⋂ C ) / P( C ) 

     = P( A | C )                                  [Proved]

 

 THEOREM - 6:

         For any 3 events A , B , C defined on the sample space S such that B ⊂ C and P( A ) > 0 , P( C | A )  ≥ P( B | A ).

 PROOF:

      P( C | A ) = P( C ⋂ A ) / P( A )      [ By definition of conditional probability]

     = [ P( B ⋂ C ⋂ A ) ⋃ P( B' ⋂ C ⋂ A ) ] / P( A )

     =  [ P( B ⋂ C ⋂ A ) ] / P( A )  +  [ P( B' ⋂ C ⋂ A ) ] / P( A )

     =    P [ ( B ⋂ C | A ) + ( B' ⋂ C | A ) ]

    Since B ⊂ C ⇒ B ⋂ C = B

           ⇒ P( C | A ) = P( B | A ) + P( B' ⋂ C | A )

           ⇒ P( C | A ) ≥ P( B |  A ).                                         [Proved]

 Pair-wise Independent Events:

         Definition:  A set of events  A₁ , A₂  , ... , A_n  are said to be pair-wise independent  if  

          P(  A_i ⋂  A_j ) = P(  A_i ) . P(  A_j )  ∀ i ≠ j

 Conditions for Mutual Independence of ' n ' Events :

    Let S denote the sample space for a number of events . The events in S are said to be mutually independent if the probability of the simultaneous occurrence of (any) finite number of them is equal to the product of their separate probabilities.

               If  A₁ , A₂  , ... , A_n  are ' n ' events , then their mutual independence , we should have  

       (i)  P(  A_i ⋂  A_j ) = P(  A_i ) . P(  A_j ) ,    [ i ≠ j ; i , j = 1,2,...,n]

      (ii) P(  A_i  ⋂  A_j  ⋂  A_k ) = P( A_i ) . P( A_j ) . P( A_k ) ,  [ i ≠ j ≠ k ; i ,      j , k = 1,2,...,n]

                                    .                                     .

                                    .                                     .

                                    .                                     .

 P( A₁ ⋂  A₂ ⋂ ... ⋂ A_n ) = P( A₁ ) . P( A₂ ) . ... . P( A_n )

 

Remarks : 

       1. It may be observed that pair-wise or mutual independence of events A₁ , A₂  , ... , A_n  is defined only when P(  A₁ ) ≠ 0 , for i = 1,2,...,n .

       2. If the events A and B are such that P( A )  ≠ 0 , P( B ) ≠ 0 and A independent of B , then B is independent of A.

      Proof :  We are given that , P( A | B) = P( A )       

                   [∵ A is independent of B ]

                   ⇒ P( A ⋂ B ) / P( B ) = P( A )

                   ⇒  P( A ⋂ B ) = P( A ) . P( B )

         ⇒ P( B ⋂ A ) / P( A ) = P( B )    [∵ P( A ) ≠ 0 ,                    A ⋂ B =  B ⋂ A ]

         ⇒ P( B | A ) = P( B )