A system consists of components 1, 2, 3, and 4 that work with probability p1, p2, p3, and p4 respectively. The signal can only pass through a component if it is working. The components are independent of each other. The system works if a signal can pass from S to E. What is the probability that the system works? Image of system architecture
I know that the probability is $P(\text{system works})= P( (1 \cap 4) \cup (1 \cap 3) \cup (2 \cap 3 \cap 4) \cup 2) = P( (1 \cap 4) \cup (1 \cap 3) \cup 2)$. Using the inclusion–exclusion principle, this probability can be calculated. However, if the diagram gets more complicated its quite messy to calculate the probability using this approach.
This is a slightly simplified problem from the chapter about conditional probabilities from Ross' textbook "Introduction to Probability models". Thus, I assume there is a simpler way to calculate the above probability using conditioning. So my question is if and how the probability of the system working can be calculated using conditional probabilities.