The rapid rise in the grid integration of low-carbon technologies, e.g. renewable energy sources (RESs) and plug-in electric vehicles (PEVs), has led to several challenges in distribution networks (DNs). This is due to the intermittent generation of RESs and uncertain loads of PEVs, both of which necessitate enhancing the flexibility requirements at the distribution level so as to accommodate the high penetration of these clean technologies in the future. To address such issues, this study proposes a mixed-integer linear programming-based expansion planning model for DNs considering the impact of high RES and PEV penetration, the associated uncertainties, and providing flexibility requirements at the distribution level. In this respect, the spherical simplex unscented transformation, an analytical uncertainty modelling method, is implemented in the planning model to take into account the forecasting errors of the uncertain green technologies. Also, in order to estimate the electric vehicle parking lot demand at each load node of the network, a new approach for PEV-charging model is suggested. To investigate the effectiveness and efficiency of the proposed probabilistic planning model, it is implemented on two test DNs, and the obtained results are thoroughly discussed.