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Probabilistic load flow (PLF) modeling is gaining renewed popularity as power grid complexity increases due to growth in intermittent renewable energy generation and unpredictable probabilistic loads such as plug-in hybrid electric vehicles (PEVs). In PLF analysis of grid design, operation and optimization, mathematically correct and accurate predictions of probability tail regions are required. In this paper, probability theory is used to solve electrical grid power load flow. The method applies two Maximum Entropy (ME) methods and a Gram-Charlier (GC) expansion to generate voltage magnitude, voltage angle and power flow probability density functions (PDFs) based on cumulant arithmetic treatment of linearized power flow equations. Systematic ME and GC parameter tuning effects on solution accuracy and performance is reported relative to converged deterministic Monte Carlo (MC) results. Comparing ME and GC results versus MC techniques demonstrates that ME methods are superior to the GC methods used in historical literature, and tens of thousands of MC iterations are required to reconstitute statistically accurate PDF tail regions. Direct probabilistic solution methods with ME PDF reconstructions are therefore proposed as mathematically correct, statistically accurate and computationally efficient methods that could be applied in the load flow analysis of large-scale networks.