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?