Reliability of safety-critical systems is a paramount issue in system engineering because in most practical situations the reliability of a non series-parallel network system has to be calculated. Some methods for calculating reliability use the probability principle of inclusion-exclusion. When dealing with complex networks, this leads to very long mathematical expressions which are usually computationally very expensive to calculate. In this paper, we provide a new expression to simplify the probability principle of inclusion-exclusion formula for intersections of unions which appear when calculating reliability on non series-parallel network systems. This new expression exploits the presence of many repeated events and has many fewer terms, which significantly reduces the computational cost. We also show that the general form of the probability principle of inclusion-exclusion formula has a double exponential complexity, whereas the simplified form has only an exponential complexity with a linear exponent. Finally, we compare its computational efficiency against the sum of disjoint products method KDH88 for a simple artificial example and for a door management system, which is a safety-critical system in aircraft engineering.