The aim of this paper is to present an overview of both the deterministic and stochastic aspects of nuclear power plant (NPP) start-up and start-up accident scenarios. The neutron kinetics and corresponding nuclear reactor control mechanisms associated with deterministic NPP start-up are explained. The stochastic behaviour of neutron kinetics is explained and the procedures required to safely start-up an NPP due to these stochastic phenomena are outlined. Specific attention is given to the stochastic behaviour of neutrons in a low neutron source regime and the corresponding mathematical methods required for modelling such phenomena during NPP start-up accidents. The Pál-Bell equations are compared against the forward generating function equations utilising the gamma approximation in their ability to model NPP start-up accidents. The Pál-Bell equations are shown to be necessary to accurately calculate the maturity time for systems with either a very low neutron source strength or a very rapid reactivity insertion profile. Both the Hurwitz calculation route and MacMillan calculation route for low neutron source systems are explained. Full deterministic and stochastic calculations are performed for multiple nuclear reactor systems to illustrate the necessary analyses required for low neutron source NPP start-up accident scenarios.