Probability, that when we send a $0$ down the network we will get back a $0$ We can send a $0$ or a $1$ over a network of $1,2...$ nodes. Unfortunately on each node with probability $p$ the message is not made different, and with probability $1-p$ the message is XOR'ed. Find recurrence relation that determines with what probability if we send a $0$ over a network of $n$ nodes we will get $0$ after $n$'th node.
Base cases.
Obviously $$a_1=p$$ and $$a_2=p^2 + (1-p)^2$$ Because we can either succeed two times or fail two times and be ok. But what about the rest?
$$a_3=p^3+{{3}\choose{2}}(1-p)^2p$$ $$a_4=p^4+ (1-p)^4+{{4}\choose{2}}(1-p)^2p^2$$ And I fail in seeing a reccurent relation between the subsequent $a_n$'s.

I assume when you say XOR, it is XOR with 1.  Take the case that you send 0. And after node $n$, call the value $x_n$.  Then 
$$\begin{align}
\text{Prob}(x_n=0) &= \text{Prob}(x_{n-1} = 0)*p + \text{Prob}(x_{n-1} = 1)*(1-p) \\
&= \text{Prob}(x_{n-1} = 0)*p + (1-\text{Prob}(x_{n-1} = 0))*(1-p) \\
&= (2p-1)\text{Prob}(x_{n-1} =0) + 1 - p
\end{align}
$$
You can figure out the rest.
