Cool, thanks for pointing that out. The field dealing with this sort of thing is called "reliability engineering", and it is apperently not widely studied.

The correct way to estimate failure rates and such is to draw a directed acyclic graph of all the components in the system so that there is at least one path from "not working" to "working". If the failure probability of each component is known, then you do the big probability join over all possible detailed scenarios for success.

Usually, the graph degenerates to one big conjunction, but if you have redundant components or systems, it can be more complex.