The future energy system embraces growing flexible demand and generation, which bring large-scale uncertainties and challenges to current deterministic network pricing methods. This paper proposes a novel reliability-based probabilistic network pricing method considering demand uncertainty. Network reliability performance, including probabilistic contingency power flow (PCPF) and tolerance loss of load (TLoL), are used to assess the impact of demand uncertainty on actual network investment cost, where PCPF is formulated by the combined cumulant and series expansion. The tail value at risk (TVaR) is used to generate analytical solutions to determine network reinforcement horizons. Then, final network charges are calculated based on the core of the Long-run incremental cost (LRIC) algorithm. A 15-bus system is employed to demonstrate the proposed method. Results indicate that the pricing signal is sensitive to both demand uncertainty and network reliability, incentivising demand to reduce uncertainties. This is the first-ever network pricing method that determines network investment costs considering both supply reliability and demand uncertainties. It can guide better sitting and sizing of future flexible demand in distribution systems to minimise investment costs and reduce network charges, thus enabling a more efficient system planning and cheaper integration.