  Reliability of safety-critical systems is an important issue in system
engineering and 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's formula for intersections of unions, which appear when
calculating reliability on non series parallel network systems. This new
expression has much less terms, which reduces enormously the computational
cost. We also show that the general form of the probability principle of
inclusion-exclusion's formula has double exponential complexity whereas the
simplified form has only exponential complexity with a linear exponent.
Finally, we illustrate how to use this result when calculating the reliability
of a door management system in aircraft engineering.
