Dependence of Events - Proof

Question

I've been learning for my probability exam checking some proofs in book and I found the following question I can't really answer. Any ideas or help?

Let's assume that events $A$ and $B$ are NOT independent. Is it possible that $A^c$ and $B^c$ are independent?

Answer 1

If the events $A^{c}$ and $B^{c}$ are independent, then $A$ and $B$ are independent.

Indeed, suppose that $A^{c}$ and $B^{c}$ are independent: \begin{align*} \mathbb{P}(A\cap B) & = 1 - \mathbb{P}(A^{c}\cup B^{c})\\\\\ & = 1 - \mathbb{P}(A^{c}) - \mathbb{P}(B^{c}) + \mathbb{P}(A^{c}\cap B^{c})\\\\ & = \mathbb{P}(A) - \mathbb{P}(B^{c}) + \mathbb{P}(A^{c})\mathbb{P}(B^{c})\\\\ & = \mathbb{P}(A) + \mathbb{P}(B^{c})(\mathbb{P}(A^{c}) - 1)\\\\ & = \mathbb{P}(A) - \mathbb{P}(B^{c})\mathbb{P}(A)\\\\ & = \mathbb{P}(A)(1 - \mathbb{P}(B^{c}))\\\\ & = \mathbb{P}(A)\mathbb{P}(B) \end{align*}

hence we conclude that $A$ and $B$ are independent. Consequently, the answer to your question is no.

Hopefully this helps!