  Given a directed graph, the Minimal Feedback Arc Set (FAS) problem asks for a
minimal set of arcs which, when removed, results in an acyclic graph.
Equivalently, the FAS problem asks to find an ordering of the vertices that
minimizes the number of feedback arcs. The FAS problem is considered an
algorithmic problem of central importance in discrete mathematics. Our purpose
in this paper is to consider the problem in the context of Erd\H{o}s-R\'{e}nyi
random directed graphs, denoted $D(n,p)$, in which each possible directed arc
is included with a fixed probability $p>0$. Our interest is the typical ratio
of the number of feedforward arcs to the number of feedback arcs that are
removed in the FAS problem. We show that as the number $n$ of vertices goes to
infinity the probability that this ratio is greater than $1+\epsilon$ for any
fixed $\epsilon > 0$ approaches zero. Similarly, letting $p$ go to zero as
$n\rightarrow \infty$ this result remains true if $p>C\log{n}/n$ where $C$
depends on $\epsilon$.
