The prediction of the reliability of laser-induced ignition holds paramount importance in the design of rocket engines. From a numerical standpoint, capturing the stochasticity of ignition for non-premixed gaseous methane and oxygen would require a wide range of variability and a substantial number of high-fidelity simulations. In this study, we perform a limited set of scale-resolving simulations designed to encompass both ignition and non-ignition events. The primary objective of our analysis is to gain insights into the underlying physical mechanisms that contribute to either success or failure and to validate against experiments. Furthermore, we conduct examinations to assess the appropriateness of initial conditions in reproducing the inherently non-deterministic nature of the experiments. Subsequently, we put forth considerations regarding multi-fidelity models, aiming to provide guidance on generating a probability map of ignition from a numerical perspective.