Mean number of edges in a hypercube graph in which random nodes are deleted

Question

I have the following problem. Suppose we have a hypercube of dimension L. Then the number of nodes is given by $2^L$ and the number of edges is given by $2^{L-1}*L$. Now each node is deleted with a probability given by $(1-p)$. The question is, what is the average number of edges for given $L$ and $p$?

Answer 1

Hint. For each edge, the probability that that edge is still there is $p^2$.

Now use additivity of expectations.

Answer 2

One idea which doesn't provide an exact answer but gives a simple recursion is to solve for the base case, L=2. Let us say this is E(2,p). Then the L=3 case is two copies of the L=2 case with edges between corresponding vertices. The expected number of edges in the two copies of the L=2 case is simply twice the expected number of edges in the L=2 case, and then looking at the other 4 edges, they are indeed independent, and each has a probability of p^2, so the expected number of edges would be E(3,p) = 2*E(2,p)+$4p^2$. Then E(n+1,p) = 2*E(n,p) + $2^{n-1}p^2$. Not sure if this is right or even what you're looking for.