Problem $2$: $(10)$

Given: A landfill system for a hazardous pollutant is lined successively with three different components,

namely, $(1)$ geotextile grade A$,(2)$ geotextile grade B$,$ and $(3)$ compacted clay as illustrated in Figure

$13\mathrm{.}23$a of the textbook and below. This arrangement could be thought of as three links in a serial

arrangement $($Figure $13\mathrm{.}23$b$)$ where a failure in any lining component disables the entire system. Assume

that component failure is not time$-$dependent and, the probabilities of failure in the linings at any given

time are as follows: $P_f(S1)=0\mathrm{.}03$;$P_f(S2)=0\mathrm{.}08$; and $P_f(S3)=0\mathrm{.}01.$

Find: If the performance of each lining is independent of the others, find the probability that the lining

system fails.