The time, \(T\) days, before an electrical component develops a fault has distribution function F given by
\[\mathrm{F}(t)=\left\{\begin{array}{ll}
1-\mathrm{e}^{-a t} & t \geqslant 0 \\
0 & \text { otherwise }
\end{array}\right.\]
where \(a\) is a positive constant. The mean value of \(T\) is 200 .
(i) Write down the value of \(a\).
(ii) Find the probability that an electrical component of this type develops a fault in less than 150 days.

A piece of equipment contains \(n\) of these components, which develop faults independently of each other. The probability that, after 150 days, at least one of the \(n\) components has not developed a fault is greater than 0.99 .
(iii) Find the smallest possible value of \(n\).