Supersaturation for hereditary properties

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$\mathrm{.}Itisknownthatthereexists$c = c(p,$\mathcal{F}$)$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from $\mathcal{F}$ is $2^{(-c+o(1))\left(\genfrac{}{}{0px}{}{n}{r}\right)}$. Let each graph in $\mathcal{F}$ have at least $t$ vertices. We show that in fact for every $ϵ>0$, there exists $\delta =\delta (ϵ,p,\mathcal{F})>0$ such that the probability of a random $r$-graph in $G(n,p)$ containing less than $\delta n^t$ induced subgraphs each lying in $\mathcal{F}$ is at most $2^{(-c+ϵ)\left(\genfrac{}{}{0px}{}{n}{r}\right)}$. This statement is an analogue for hereditary properties of the supersaturation theorem of Erd\H{o}s and Simonovits. In our applications we answer a question of Bollob'as and Nikiforov.