Answer
3 parallel subsystems would be required to guarantee that the system would operate properly at least 99% of the time.
Work Step by Step
Trial assuming 3 subsystems-
Event A: Subsystem 1 works.
Event B: Subsystem 2 works.
Event C: Subsystem 3 works.
Therefore,
$P$(any subsystem in system 3 works) = $P$(A or B or C) =$P$(A U B U C).
$P$(A U B U C) = $P$(A)+$P$(B)+$P$(C)-$P$(A∩B)-$P$(B∩C)-$P$(A∩C)+$P$(A∩B∩C)
$P$(A U B U C) = 0.81+0.81+0.81-0.6561-0.6561-0.6561+($P$(A)*$P$(B)*$P$(C))
$P$(A U B U C) = $0.81+0.81+0.81-0.6561-0.6561-0.6561+(0.81*0.81*0.81)$
$P$(A U B U C) = $2.43-1.9683+0.531441$.
$P$(A U B U C) = 0.993141 = $99.3141$%.
