Riddles, brain puzzles and mathematical problems

[BAndrew]

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.

As I said I haven't been taught about possibilities yet at school so I am not exactly sure, but I think
P(A U B) = P(A) + P(B) - P(A+B) *

*P(A+B) is symbolized with an inverted U