  Given a $k$-uniform hypergraph $\mathcal{H}$ and sufficiently large $m \gg
m_0(\mathcal{H})$, we show that an $m$-element set $I \subseteq
V(\mathcal{H})$, chosen uniformly at random, with probability $1 -
e^{-\omega(m)}$ is either not independent or is contained in an
almost-independent set in $\mathcal{H}$ which, crucially, can be constructed
from carefully chosen $o(m)$ vertices of $I$. As a corollary, this implies that
if the largest almost-independent set in $\mathcal{H}$ is of size
$o(v(\mathcal{H}))$ then $I$ itself is an independent set with probability
$e^{-\omega(m)}$. More generally, $I$ is very likely to inherit structural
properties of almost-independent sets in $\mathcal{H}$.
  The value $m_0(\mathcal{H})$ coincides with that for which Janson's
inequality gives that $I$ is independent with probability at most
$e^{-\Theta(m_0)}$. On the one hand, our result is a significant strengthening
of Janson's inequality in the range $m \gg m_0$. On the other hand, it can be
seen as a probabilistic variant of hypergraph container theorems, developed by
Balogh, Morris and Samotij and, independently, by Saxton and Thomason. While
being strictly weaker than the original container theorems in the sense that it
does not apply to all independent sets of size $m$, it is nonetheless
sufficient for many applications and admits a short proof using probabilistic
ideas.
