According to this answer, a random graph on $n$ vertices is a graph which has each of the $n\choose2$ edges independently with probability $1/2$ each. The probability of at most $3n-6$ edges (which is a necessary condition for planarity) is: $$2^{-n(n-1)/2}\sum_{k=0}^{3n-6}{n(n-1)/2\choose k}$$
Therefore, the plot for this equation is:
If I choose a random graph with 10 vertices, what is the probability that it will remain planar after adding an edge between two randomly chosen vertices?
To be more clear:
Consider event $P_{1}$, the event of choosing a planar graph $G_{1}$ in the universe $U_{10}$ of graphs with 10 vertices.
Consider a subset $S$ of the universe of graphs with 11 vertices $U_{11}$ conditionally defined by the graph $G_{1}$ we got in event $P_{1}$ in this way: $S$ is formed by all possible graphs that can result from adding one edge to $G_{1}$.
And then finally consider event $P_{2}$, the event of choosing a planar graph in the universe $S$ defined in the previous step.
Therefore I want to know what is the probability of event $P_{2}$.
