Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of $μ$. Łuczak and Seierstad (2009, Random Structures & Algorithms, 35(3), 271--293) showed that, as $μ\to -\infty$, the strongly connected components of a random digraph with $n$ vertices and $m = n(1 + μn^{-1/3})$ directed edges are, with high probability, only isolated vertices and cycles. We call such digraphs elementary digraphs. We express the probability that a random digraph is elementary as a function of $μ$. Those results are obtained using techniques from analytic combinatorics, developed in particular to study random graphs.