  Let $\mathcal{H}$ denote a collection of subsets of $\{1,2,\ldots,n\}$, and
assign independent random variables uniformly distributed over $[0,1]$ to the
$n$ elements. Declare an element $p$-present if its corresponding value is at
most $p$. In this paper, we quantify how much the observation of the
$r$-present ($r>p$) set of elements affects the probability that the set of
$p$-present elements is contained in $\mathcal{H}$. In the context of
percolation, we find that this question is closely linked to the near-critical
regime. As a consequence, we show that for every $r>1/2$, bond percolation on
the subgraph of the square lattice given by the set of $r$-present edges is
almost surely noise sensitive at criticality, thus generalizing a result due to
Benjamini, Kalai and Schramm.
