  For a compact surface $S$ with constant negative curvature $-\kappa$ (for
some $\kappa>0$) and genus $g\geq2$, we show that the tails of the distribution
of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the
intersection number of the closed geodesics and $l(\cdot)$ denotes the
geometric length) are estimated by a decreasing exponential function. As a
consequence, we find the asymptotic normalized average of the intersection
numbers of pairs of closed geodesics on $S$. In addition, we prove that the
size of the sets of geodesics whose $T$-self-intersection number is not close
to $\kappa T^2/(2\pi^2(g-1))$ is also estimated by a decreasing exponential
function. And, as a corollary of the latter, we obtain a result of S. Lalley
which states that most of the closed geodesics $\alpha$ on $S$ of length $\leq
T$ have roughly $\kappa l(\alpha)^2/(2\pi^2(g-1))$ self-intersections, when $T$
is large.
