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.