  Given a graph $G$, we consider the model where $G$ is given a random
orientation by giving each edge a random direction. It is proven that for
$a,b,s\in V(G)$, the events $\{s\to a\}$ and $\{s\to b\}$ are positively
correlated. This correlation persists, perhaps unexpectedly, also if we first
condition on $\{s\nto t\}$ for any vertex $t\neq s$. With this conditioning it
is also true that $\{s\to b\}$ and $\{a\to t\}$ are negatively correlated.
  A concept of increasing events in random orientations is defined and a
general inequality corresponding to Harris inequality is given.
  The results are obtained by combining a very useful lemma by Colin McDiarmid
which relates random orientations with edge percolation, with results by van
den Berg, H\"aggstr\"om, Kahn on correlation inequalities for edge percolation.
  The results are true also for another model of randomly directed graphs.
