  We establish mild conditions under which a possibly irregular, sparse graph
$G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient
each edge in either direction with probability $1/2$ independently. We show
that if $G$ satisfies a minimum degree condition of $(1+c_1)\log_2{n}$ and has
Cheeger constant at least $c_2\frac{\log_2\log_2{n}}{\log_2{n}}$, then the
resulting randomly oriented directed graph is strongly connected with high
probability. This Cheeger constant bound can be replaced by an analogous
spectral condition via the Cheeger inequality. Additionally, we provide an
explicit construction to show our minimum degree condition is tight while the
Cheeger constant bound is tight up to a $\log_2\log_2{n}$ factor.
