Riddles, brain puzzles and mathematical problems

[Oscar House]

It doesn't make any sense to subtract it though, because it's another possibility. In other words, there are three ways in which a current can pass through the circuit:

1) Via the top half (two switches)
2) Via the bottom half (three switches)
3) Via both (five switches)

Why is the solution not (2/3)^3+(2/3)^2+(2/3)^5? I honestly want to know, I have to take a final on this stuff next month and I find probability so vague and indirect.

I think it has less to do with probability and more to do with the way electricity works. The current loops back to where it's coming from, causing a short circuit.