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.