For a compact surface S with constant curvature −κ (for some κ > 0) and genus g ≥ 2, we show that the tails of the distribution of the normalized intersection numbers 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 average of the normalized intersection numbers of pairs of closed geodesics on S. In addition, we prove that

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... he size of the sets of geodesic arcs whose T -self-intersection number is not close to κT 2 /(2π 2 (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 α on S with l(α) ≤ T have roughly κl(α) 2 /(2π 2 (g−1)) self-intersections, when T is large.