If I have a simple graph $G$ with $n$ vertices and $m$ edges, then I want to create a randomly induced graph $G_x$ by selecting vertices with a probability of $n/2m$. The edges of $G_x$ are defined to be all $e\in E(G)$ between $u$ and $v$ where vertices $u, v \in V(G_x)$. How can I find the expected number of edges in $G_x$?
I started by calculating the expected number of vertices in $G_x$ as $n^2/2m$ and so the expected number of edges should be the probability that both $u,v\in G_x$ and $uv \in E(G)$. But this expected value is difficult to calculate. Can I have some suggestions without explicit answers?
As explained in a comment, the expected number of edges is
$$ \left(\frac n{2m}\right)^2m=\frac{n^2}{4m}\;. $$
For linearity of expectation, see e.g. Wikipedia. The main thing to know about it is that it does not require independence. That's what makes it so useful – you can often easily calculate the expected value using the linearity of expectation when calculating the entire distribution and then using it to calculate the expected value would be very complicated due to correlations (as in the present case).