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Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a randombinomial subset of $N$ where every element of $N$ is taken to belong to thesubset 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 allminimal 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 adecreasing function, which implies a uniform bound on the coarseness of thethresholds 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 momentmethod gives a good approximation of its measure, or a subfamily which can bewell approximated by a family with all minimal sets of size strictly smallerthan $k$.  Finally, we relate the (fractional) expectation threshold and the probabilitythreshold of such a family, using duality of linear programming. This isrelated to the threshold conjecture of Kahn and Kalai.

Thresholds and expectation-thresholds of monotone properties with small minterms