Answer
5 components.
Work Step by Step
The event "the system will succeed" is the complement of "the system will not succeed". So, :
$P(the~system~will~succeed)=1-P(the~system~will~not~succeed)$
We want that:
$P(the~system~will~succeed)\gt0.9999$
$1-P(the~system~will~not~succeed)\gt0.9999$
$-P(the~system~will~not~succeed)\gt0.9999-1$
$-P(the~system~will~not~succeed)\gt-0.0001$
$P(the~system~will~not~succeed)\lt0.0001$
- 2 components fail (a):
$P(two~components~fail)=0.0225\gt0.0001$
- 3 components fail (a):
$P(3~components~fail)=P(component~1~fail)\times P(component~2~fail)\times P(component~3~fail)=0.15\times0.15\times0.15=0.003375\gt0.0001$
- 4 components fail (a):
$P(4~components~fail)=P(component~1~fail)\times P(component~2~fail)\times P(component~3~fail)\times P(component~4~fail)=0.15\times0.15\times0.15\times0.15=0.00050625\gt0.0001$
- 5 components fail (a):
$P(5~components~fail)=P(component~1~fail)\times P(component~2~fail)\times P(component~3~fail)\times P(component~4~fail)\times P(component~5~fail)=0.15\times0.15\times0.15\times0.15\times0.15=0.0000759375\lt0.0001$
