A and B be two events of the same probability space. Equivalent

Question

I have this probation calculation demo exercise and I do not know very well

Let $A$ and $B$ be two events of the same probability space. They are equivalent

$A$ and $B$ are independent

$A$ and $B^\complement$ are independent

$A^\complement$ and $B$ are independent

$A^\complement$ and $B^\complement$ are independent

((being c complemented))

Apart from the question I do not understand very wellI have managed to show that $A^\complement$ and $B^\complement$ are independent if and only if $A$ and $B$ are independent, but for this new exercise I can not use any of this obtained. Can someone give me a hand how to continue? Is there a rule or something that you do not know that can be applied to prove it?

Answer 1

Let it be that $A$ and $B$ are independent.

Then:$$P(A\cap B^{\complement})=P(A)-P(A\cap B)=P(A)-P(A)P(B)=P(A)(1-P(B))=P(A)P(B^{\complement})$$

showing that also $A$ and $B^{\complement}$ are independent.

Applying this principle it can be shown that the $4$ statements in your question are equivalent.

That's what this is about.