  We asymptotically estimate the variance for the distribution of closed
geodesics in small random balls or annuli on the modular surface
$\Gamma\backslash\mathbb{H}$. A probabilistic model in which closed geodesics
are modeled using random geodesic segments is proposed, and we rigorously
analyze this model using mixing of the geodesic flow in
$\Gamma\backslash\mathbb{H}$. This leads to a conjecture for the asymptotic
behavior of the variance, which unlike in previously explored cases is not
equal to the expected value. We prove this conjecture for small balls and
annuli, resolving a question left open by Humphries and Radziwill.
