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Reliability can be defined as the probability that a system will run without failure for a given period of time.
A system is usually composed of individual units. We can define an event $U_i$ representing the case where the $i^{th}$ unit is up (i.e. not down / failing). Furthermore, we assume that whether an individual unit is up or down is independent of whether other units are up or down.
Similarly we can define an event $F_i$ to represent the case where the $i^{th}$ unit is down. Note that $\mathbb{P}(F_i) = 1 - \mathbb{P}(U_i)$.
In the below examples, for convenience we will use the following shorthand notation:
Examples
Series circuit
Consider the following system:
The calculate the probability the system is up (i.e. there exists a path from left to right where we only pass through "up" units):
Parallel circuit
Next consider the following system:
The calculate the probability the system is up:
Here we make use of De Morgan's laws to convert from $\mathbb{P}(U_1\cup{U_2}\cup{U_3})$ to $1 - \mathbb{P}(F_1\cap {F_2}\cap {F_3})$, making use of the fact that $U_i^C = F_i$.
