Question

设A、B、C三个事件相互独立,求证：A∪B、AB、A-B皆与C独立.

Answer 1

1°证明：P（AB）=P（A）*P(B);P（AC）=P（A）*P(C);P（BC）=P（B）*P(C).

P(ABC)=P(A)*P(B)*P(C)=P(AB)*P(C)

P(A∪B)=P(A)+P(B)-P(AB) P(A∪B)*P(C)=P(A)*P(C)+P(B)P(C)-P(AB)P(C)=P(AC)+P(BC)-P(ABC)=P[(A∪B)C]

P(A-B)*P(C)=P(A)P(C)-P(AB)P(C)=P(AC)-P(ABC)=P(AC-ABC)=P((A-B)C)