This paper proposes a statistical approach for section-based fault location in medium voltage (MV) grids with underground cables, using Bayesian inference. The proposed approach considers several important uncertainties in the MV grid, including measurement errors, fault breakdown resistance, and the inaccuracies of zero-sequence parameters. The approach first obtains the prior distribution of the fault position from the component failure database, the readings of the transmitted fault indicators, and the relevant digging activity record. With the estimated prefault grid status and the measured transient voltages/currents, the posterior distribution is then calculated based on Bayes' theorem. To solve the problem numerically, the Monte Carlo integration is applied and a two-step calculation procedure is proposed. Simulations are performed on a typical European MV feeder to demonstrate the feasibility of the approach. The distribution grid operators can use the calculated posterior distribution to rank the possible faulted sections and to facilitate the restoration process, which can reduce the interruption duration of power supply.