I want to approximate the properties of a given, deterministic, undirected Graph $G$ without multiple edges. To be precise: I want to know the probability $P$ that $G$ with $n$ edges and $z$ vertices contains a clique $K_x$ of size $x$.
My intuition is, that probability of two vertices being connected by an edge in a random graph is equivalent to the density $den(G)$ of a given, deterministic Graph, hence i could use the following formula from the book "A Guide to Graph Colouring" by R. Lewis (2015), formulated for random graphs, $p$ being the probability that two vertices are connected by an edge:
$P(\exists K_x \subseteq G) = 1-(1-p^{\binom{x}{2}})^{\binom{n}{x}}$
and substitute $p$ with the $den(G) = \dfrac{n}{\frac{m(m-1)}{2}}$, $m$ being the number of edges of a complete Graph $K_z$
Is that intuition correct? If so, how can i argue that this is correct or even proof it?
Edited according to comments to make my question more clear.