Significant efforts are under way to develop innovative ignition systems for spark-ignition engines used in transportation. Within this context, passive pre-chamber technology has emerged as a promising alternative for passenger cars. However, several uncertainties remain regarding the operation of this concept at low engine loads and speeds, as well as the impact of specific design features on combustion stability. Previous investigations have indicated that the tangential angle of the pre-chamber holes can play a vital role in stabilizing the combustion process. Nonetheless, the underlying thermo-physical phenomena responsible for these results have not yet been thoroughly studied. To address these knowledge gaps, this paper presents a numerical study using a computational fluid dynamics model that has been validated with experimental results. An alternative modeling methodology was developed to conduct multi-cycle large-eddy simulations and investigate two different pre-chamber configurations, one with tangential holes and the other with radial holes. The results revealed an intriguing correlation between the combustion stability and the spatial distribution of the flame inside the pre-chamber. The cycle-to-cycle dispersion of pre-chamber flow variables was significantly higher when using radial holes compared to tangential holes, potentially explaining the unstable behavior of the former design. Additionally, the undesirable flow-field of the radial-hole pre-chamber caused the flame to evolve asymmetrically, resulting in substantial variations in the ejected jets. This asymmetry can significantly affect the morphology of the main chamber ignition in each cycle.