Probabilistic risk assessment (PRA) is used as an essential tool for risk-informed decision-making in the nuclear industry. The fault and event trees play a crucial role in PRA to estimate the probability of system failure based on the failure probabilities of components. The fault trees or event trees for an actual power plant unit can be fairly large in size with several different types of logic gates, interconnected events, dependent events, etc. A large fault tree can include hundreds of gates, basic events (BEs), multiple occurring events (MOEs), and dependent events. Complex connectivities can give rise to excessive computational demand and storage requirements for the analysis. Fault and event trees can be solved using the minimal cut-set approaches, or advanced quantification techniques such as Binary decision diagrams or Bayesian networks. However, these techniques can be computationally inefficient for larger fault trees and can run out of memory/storage space. This study focuses on developing and proposing a new approach for accurate estimation of the system-level risk while improving the computational efficiency significantly. More specifically, an attempt is made to reduce the complexity of the analysis of MOEs and dependent events in fault trees. The proposed algorithms in this study present a significant improvement over traditional approaches which makes it highly promising for additional development. The computational efficiency of the proposed approach over the traditional approach is illustrated for fault trees with a varying number of events and different types of logic gate connections.