For a compact surface S with constant negative curvature −κ (for some κ>0) and genus g≥2, we show that the tails of the distribution of i(α,β)/l(α)l(β) (where i(α,β) is the intersection number of the closed geodesics α and β and l(⋅) 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 κT2/(2π2(g−1)) is also estimated by a decreasing exponential function. And, as a corollary of the latter, we obtain a result of Lalley which states that most of the closed geodesics γ on S with l(γ)≤T have roughly κl(γ)2/(2π2(g−1)) self-intersections, when T is large.