I have 3 similar problems but I don't quite understand the differences between them and how to approach each.
First 2 Problems
Lets say you have 2 circuits where all relays function independently:
I'll call the top 1 circuit 1 and the bottom 1 circuit 2. Each of them has a flow going from A to B, and has a probability of closing the $i$th relay with a probability of $P_i$ where $i=1,2,3,4...$
I was working on these problems and the answers the book gave me for the probability of the circuit flows being completed were that for circuit 1 the probability was $P_{1}P_{3} + P_{1}P_{4} + P_{2}P_{3} + P_{2}P_{4}$ and for circuit 2 it was $P_{5}(P_{1}P_{2} + P_{3}P_{4} - P_{1}P_{2}P_{3}P_{4})$
I don't under stand why in the first case you don't have to subtract out the product of all 4 probabilites of the relays like you do in the second circuit.
3rd Problem:
An engineering system consists of $n$ components is said to be a k out of n system (where k is less than n) if the system functions if and only if at least k of the n components function. Suppose that all components function independently of each other. If the $i$th component functions with probability $P_{i}$ where $i=1,2,3,4$ what is the probably that a 2 out of 4 system functions?
For this one then I'm confused as to whether I can just add up all the combinations of pairs of components and add them together like the first circuit, or I have to subtract out overlaps like the second circuit.
I guess I'm really not understanding how the inclusion-exclusion rule is working in these problems. Thanks for any help.