For two fixed graphs $T$ and $H$, a positive integer $n$ and a real number $p$ in $[0, 1]$ let $ex(G(n, p), T, H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n, p)$. In this talk we discuss this variable, its phase transition as a function of $p$ and its connection to the deterministic function counting the maximum number of copies of $T$ in an $H$-free graph on $n$ vertices. Based on joint works with N. Alon, A. Kostochka and W. Samotij.