We consider a \emph{family} $(P_ω)_{ω\in Ω}$ of elliptic second order differential operators on a domain $U_0 \subset \mathbb{R}^m$ whose coefficients depend on the space variable $x \in U_0$ and on $ω\in Ω,$ a probability space. We allow the coefficients $a_{ij}$ of $P_ω$ to have jumps over a fixed interface $Γ\subset U_0$ (independent of $ω\in Ω$). We obtain polynomial in the norms of the coefficients estimates on the norm of the solution $u_ω$ to the equation $P_ωu_ω= f$ with transmission and mixed boundary conditions (we consider ``sign-changing'' problems as well). In particular, we show that, if $f$ and the coefficients $a_{ij}$ are smooth enough and follow a log-normal-type distribution, then the map $Ω\ni ω\to \|u_ω\|_{H^{k+1}(U_0)}$ is in $L^p(Ω)$, for all $1 \le p < \infty$. The same is true for the norms of the inverses of the resulting operators. We expect our estimates to be useful in Uncertainty Quantification.