Reliability Probability problem What is the Probability that at least one close path is formed from $A$ to $B$ where  each switch has a probability of close $p$ and each switch is mutually independent of each other?  

Proposed Solution
Let event $A$ be such that
$$\begin{align}
A &= \{\text{Current Flows From A to B}\}  \\
&= \{ S_0^c  S_1^c  S_2^c  S_3^c, S_0^o  S_1^o S_2^c  S_3^c , S_0^c  S_1^o S_2^c S_3^c, S_0^o  S_1^c  S_2^c  S_3^c, S_0^c  S_1^c S_2^o S_3^c \} \\
\end{align}$$
where $S_0^c$ or $S_0^o$  shows the switch $S_0$ Close or Open respectively. Now the probability of $A$ is
$$\begin{align}
p[A] &= p^4 + p^2(1-p)^2 + p^3(1-p) + p^3(1-p) + p^3(1-p) \\
&= p^2(1+pq)\\
\end{align}$$
where  $q = 1-p$.

Notice that 
$$\begin{align}
p[A] &= p^4 + p^2(1-p)^2 + p^3(1-p) + p^3(1-p) + p^3(1-p) \\
&= p^4+3p^3(1-p)+p^2(1-p)^2\\
&= -p^4+p^3+p^2
\end{align}$$
where  $q = 1-p$ and $-p^4+p^3+p^2=p^2(1+pq)$ so as others nocited: your calculation for the event probability is correct.
