(In)dependence of pairs of events I'm currently working through my probability homework, and one of the questions is:
Two events are independent of each other when $P(A \cap B) = P(A) \cdot P(B)$. What can you say about the (in)dependence of the following pairs of events:
$$
\begin{align*}
&(a)\textrm{ } A \textrm{ and } \neg B\\
&(b)\textrm{ } \neg A \textrm{ and } B\\
&(c)\textrm{ } \neg A \textrm{ and } \neg B
\end{align*}
$$
What can you say about this? I'm not exactly sure what to do.
This is what I've managed now:
(a)
\begin{align*}
P(A \cap \neg B) &= P(A) - P(A \cap B)\\    
&= P(A) - P(A) \cdot P(B)\\
&= P(A) \cdot (1 - P(B))\\
&= P(A) \cdot P(\neg B) \textrm{, so they are independent}
\end{align*}
(b)
\begin{align*}
P(\neg A \cap B) &= P(B) - P(A \cap B)\\    
&= P(B) - P(A) \cdot P(B)\\
&= P(B) \cdot (1 - P(A))\\
&= P(\neg A) \cdot P(B) \textrm{, so they are independent}
\end{align*}
but I'm still stuck on (c).

To give you a start.
If $A$ and $B$ are independent events then:$$P(A\cap B^c)=P(A)-P(A\cap B)=P(A)-P(A)P(B)=P(A)[1-P(B)]=P(A)P(B^c)$$
In the second equality the independence is used.
