 Solution

Choose each vertex independently with probabilityand take the induced drawing. Let its numbers of vertices, edges, and crossing pairs be $N,E,T$. Since crossing edges have four distinct endpoints,Part (i), with the harmless positive constant dropped, gives $T\ge E-3N$ for every outcome. Taking expectations,For $p=4n/e$, the right side is $4n^2/e$. Dividing by $p^4=256n^4/e^4$ proves the crossing lemma estimate