  For a triangle in the hyperbolic plane, let $\alpha,\beta,\gamma$ denote the
angles opposite the sides $a,b,c$, respectively. Also, let $h$ be the height of
the altitude to side $c$. Under the assumption that $\alpha,\beta, \gamma$ can
be chosen uniformly in the interval $(0,\pi)$ and it is given that
$\alpha+\beta+\gamma<\pi$, we show that the strong triangle inequality $a + b >
c + h$ holds approximately 79\% of the time. To accomplish this, we prove a
number of theoretical results to make sure that the probability can be computed
to an arbitrary precision, and the error can be bounded.
