  We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations
on the edges. For three fixed distinct vertices s,a,b we study the correlation,
in the combined probability space, of the events a -> s and s -> b.
  For G(n,p), we prove that there is a p_c=1/2 such that for a fixed p<p_c the
correlation is negative for large enough n and for p>p_c the correlation is
positive for large enough n. We conjecture that for a fixed n\ge 27 the
correlation changes sign three times for three critical values of p.
  For G(n,m) it is similarly proved that, with $p=m/\binom{n}{2}$, there is a
critical p_c that is the solution to a certain equation and approximately equal
to 0.7993. A lemma, which computes the probability of non existence of any k
directed edges in G(n,m), is thought to be of independent interest.
  We present exact recursions to compute P(a -> s)$ and P(a -> s, s -> b)$. We
also briefly discuss the corresponding question in the quenched version of the
problem.
