  Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random
binomial subset of $N$ where every element of $N$ is taken to belong to the
subset independently with probability $p$ . This defines a product measure
$\mu_p$ on the power set of $N$, where for $\mathcal{A} \subseteq 2^N$
$\mu_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]$.
  In this paper we study upward-closed families $\mathcal{A}$ for which all
minimal sets in $\mathcal{A}$ have size at most $k$, for some positive integer
$k$. We prove that for such a family $\mu_p(\mathcal{A}) / p^k $ is a
decreasing function, which implies a uniform bound on the coarseness of the
thresholds of such families.
  We also prove a structure theorem which enables to identify in $\mathcal{A}$
either a substantial subfamily $\mathcal{A}_0$ for which the first moment
method gives a good approximation of its measure, or a subfamily which can be
well approximated by a family with all minimal sets of size strictly smaller
than $k$.
  Finally, we relate the (fractional) expectation threshold and the probability
threshold of such a family, using duality of linear programming. This is
related to the threshold conjecture of Kahn and Kalai.
