Consider a power supply system that has two components $($Net and Battery$)$ connected in a Cold$-$Standby configuration:

The primary power comes from a normal power network $($Net$).$ The Net's lifetime distribution follows an exponential distribution with a failure rate $=\frac{1}{400}$ failures$/$day;

In addition to that, a battery is kept charged and is used if necessary. The Battery's lifetime distribution follows an exponential distribution with a failure rate $=\frac{1}{50}$ failures$/$day$.$

For a service time of $100$ days, calculate the overall system reliability by assuming the switching is not perfect with a reliability of $0\mathrm{.}95$ and the standby component is instantaneously switched on when the first component fails.