Let A, B and C are independent events. How am I supposed to prove that:
A′, B′ and C′ are independent.
A, B′ and C′ , are independent.
A, B and C' are independend.
This is my approach:
for Nr 3.
P(ABC') = P(A)P(B)P(C'). But P(AB)=P(ABC)+P(ABC') and using independence P(A)P(B) = P(A)P(B)P(C)+ P(ABC'), therefore P(A)P(B)(1-P(C))=P(ABC'), P(ABC') = P(A)P(B)P(C').
for Nr 2.
P(AB'C') = P(A)P(B')P(C'). But P(AC')=P(ABC')+P(ABC') and using independence P(A)P(C') = P(A)P(B)P(C')+ P(A B' C'), therefore P(A)P(C')(1-P(B))=P(AB' C'), P(AB'C') = P(A)P(B')P(C').
And for Nr 1.
P(A'B'C') = P(A')P(B')P(C'). But P(A'B')=P(A'B' C )+P(A'B'C') and using independence P(A')P(B') = P(A')P(B')P(C)+ P(A'B'C'), therefore P(A')P(B')(1-P(C))=P(A'B'C'), P(ABC') = P(A')P(B')P(C').
What do you think people? is this way of proving right?