Recently, stochastic geometry has been applied to provide tractable performance analysis for low earth orbit (LEO) satellite networks. However, existing works mainly focus on analyzing the ``coverage probability'', which provides limited information. To provide more insights, this paper provides a more fine grained analysis on LEO satellite networks modeled by a homogeneous Poisson point process (HPPP). Specifically, the distribution and moments of the conditional coverage probability given the point process are studied. The developed analytical results can provide characterizations on LEO satellite networks, which are not available in existing literature, such as ``user fairness'' and ``what fraction of users can achieve a given transmission reliability ''. Simulation results are provided to verify the developed analysis. Numerical results show that, in a dense satellite network, {\color{black}it is} beneficial to deploy satellites at low altitude, for the sake of both coverage probability and user fairness.