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Let $G$ be a undirected graph with $n$ vertices and no edges, and let $f(k)$ be the probability that if we add $k$ edges randomly to $G$ that $G$ will be connected. How would one determine $f(k)$ for a given $n\in\mathbb N$?

Specifics:

When we add an edge to the graph $G$, any edge that can exist would be equally likely to be added. We can not add the same edge twice, so when $k=\frac{n^2+n}2$ (the number of possible edges in a graph with $n$ verticies), then $f(k)=1$.

Definition for connected may be found on wikipedia.

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A classical result of Erdős and Rényi states that if $k = \frac{n}{2} (\log n + c)$, then as $n\to\infty$, the probability that the graph is connected tends to $e^{-e^{-c}}$. You can find the proof in any textbook on random graphs.

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