Let A, B, C be three mutually independent events. Consider  the two statements S1 and S2.S1: A and B∪C are independentS2: A and B∩C are independentThen,

P[A∩(B∪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∩C)]=P(A)P(B∪C) Therefore, S1 is true. P(A∩(B∩C))=P(A)P(B)P(C)=P(A)P(B∩C) Therefore, S2 is also true.