Problem source

In a rabbit warren, underground chambers 
$A, B, C$ and $D$ are at the vertices of a square, 
and burrows join $A$ to $B$, \  $B$ to $C$, \ $C$ to $D$ and $D$ to $A$. 
Each of the chambers also has a tunnel to the surface. 
A rabbit finding itself in any chamber runs along one 
of the two burrows to a neighbouring chamber, or 
leaves the burrow through the tunnel to the surface. 
Each of these three possibilities is equally likely.
Let $p_A\,$, $p_B\,$, $p_C$ and $p_D$ be the probabilities 
of a rabbit leaving the burrow through the tunnel from chamber $A$, 
given that it is currently in chamber $A, B, C$ or $D$, respectively.
\begin{questionparts}
\item Explain why  $p_A = \frac13 + \frac13p_B + \frac13 p_D$.
\item Determine $p_A\,$.
\item Find the probability 
that a rabbit which starts in chamber $A$ does not visit chamber~$C$, 
given that it eventually leaves the burrow through the tunnel in chamber $A$.
\end{questionparts}