Probability that a random graph is planar
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 (that's a necessary condition for planarity) is $$2^{-n(n-1)/2}\sum_{k=0}^{3n-6}{n(n-1)/2\choose k}$$ There are standard ways to estimate the sum.