Let $L^\omega_{\infty\omega}$ be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let $p(n)$ be the edge probability of the random graph on $n$ vertices. It is shown that if $p(n) \ll n^{-1}$ satisfies certain simple conditions on its growth rate, then for every $\sigma\in L^\omega_{\infty\omega}$, the probability that $\sigma$ holds for the random graph on $n$ vertices converges. In fact, if $p(n) = n^{-\alpha}, \alpha > 1$, then the probability is either smaller than $2^{-n^d}$ for some $d > 0$, or it is asymptotic to $cn^{-d}$ for some $c > 0, d \geq 0$. Results on the difficulty of computing the asymptotic probability are given.