  We consider the membrane model on a box $V_N\subset \mathbb{Z}^n$ of size
$(2N+1)^n$ with zero boundary condition in the subcritical dimensions $n=2$ and
$n=3$. We show optimal estimates for the probability that the field is positive
in a subset $D_N$ of $V_N$. In particular we obtain for $D_N=V_N$ that the
probability to be positive on the entire domain is exponentially small and the
rate is of the order of the surface area $N^{n-1}$.
