Let A, B, C be three mutually independent events. Consider the two statements S1 and S2.S1 : A and B ∪ C are independent.S2 : A and B ∩ C are independent.Then

We are given that                      P(A∩B)=P(A)P(B)P(B∩C)=P(B)P(C)P(C∩A)=P(C)P(A)P(A∩B∩C)=P(A)(B)P(C)We have,              P(A∩(B∩C))=P(A∩B∩C)=P(A)P(B)P(C)=P(A)P(B∩C)⇒ A and B a C are independentTherefore, S2 is true. Also,                  P[(A∩(B∪C)]=P[(A∩B)∪(A∩C)]  =P(A∩B)+P(A∩C)−P[(A∩B)∩(A∩C)]  =P(A∩B)+P(A∩C)−P(A∩B∩C)  =P(A)P(B)+P(A)P(C)−P(A)P(B)P(C)  =P(A)[P(B)+P(C)−P(B)P(C)]  =P(A)[P(B)+P(C)−P(B∩C)]  =P(A)P(B∪C)Therefore, A and B ∪ C are independent