Replacing initial probability in $G(n,\frac{1}{2})$ with $G(n,\frac{1}{3})$ for not appearance edges

Question

I have a question; maybe so simple but practical:

In Erdos-Renyi binomial random graph $G(n,p)$; set $p=\frac{1}{2}$. So with probability $1/2$ some edges will appear and some not.

Now the question is what kind of graph will we obtain if we replace $p=\frac{1}{2}$ with $p'=\frac{1}{3}$ for not appearance edges; in other words, if in $G(n, \frac{1}{2})$ for $v_i \nsim v_j$ we draw the edge $v_i v_j $ in $G(n, \frac{1}{3})$; then by considering the union of all the edges in the mentioned graphs and assuming that these two graphs are independent; then what will be the gained graph in our initial model?

Thanks!

Answer 1

Note that the graph is distributed as $G(n, p')$ for some $p'$. Why? Clearly the appearance of an edge in the final graph is independent of the choices for the other edges. Fix an edge $e$ and so $p' = P(e \text{ appears in final graph})$. Now, note that $$ \{e \text{ appears in final graph}\} = \{e \in \text{intermediate graph}\} \cup \{e \notin \text{intermediate graph}, e \in \text{final graph}\}, $$ This is a disjoint union and these probabilities should be straightforward to calculate.