I have a $Z^2$ lattice. Every element $(z_1, z_2) \in Z^2$ is connected to one of its 4 neighbours $(z_1, z_2) + (0,1)$, $(z_1, z_2) + (0,-1)$, $(z_1, z_2) + (1,0)$, $(z_1, z_2) + (-1,0)$ by an edge. The edges can be "open" with a probability $p$ and "closed" with a probability 1-p.

I call $\sigma_n$ the total number of self avoinding walks starting from $(0, 0)$ and composed of $n$ steps through the edges. Then I define the event {$ N_n = k $} = {"the number of SAW composed of n OPEN steps is k"}. My question is: is it correct writing that the probability of this event is the following?

$$ P(N_n = k) = \binom {\sigma_n} {k} (p^n)^k (1-p^n)^{\sigma_n - k }. $$

This would mean that the events {"this SAW is open"} and {"this other SAW is open"} are all independent one from the other, but I think it should be wrong because some SAWs necessarily share some edge one with the other one. But on the other hand, this is the only way I am able to justify the last equality in the following expression, which I find in many text books of percolation theory, $$ P( N_n \geq 1) = \sum_{k=1}^{\sigma_n} P(N_n =k) \leq \sum_{k=1}^{\sigma_n} k P(N_n = k) =E[N_n] = p^n \sigma_n. $$ where E is the expectation. Could anyone help me to clarify this point?