For simplicity, assume that the successful firing of a rocket’s propulsion syste
ID: 2922158 • Letter: F
Question
For simplicity, assume that the successful firing of a rocket’s propulsion system depends on three components operating successfully; the controlling computer, the ignition system, and the fuel pump. Assume that the reliabilities of these three components on each firing of the rocket are 0.9995, 0.9997, and 0.9998, respectively.
a) If the three components fail or survive independently, what is the probability of failure of the rocket the next time it is required?
b) Assuming independence between rocket firings, what is theprobability that the rocket will successfully fire the next 100 times that it is required?
c) How does the result of part (a) change if the failure of the three components are assumed to be disjoint events, rather than independent events?
Explanation / Answer
a) Probability of failure of the rocket the next time it is required is computed as:
= 1 - Probability that all the components work perfectly without fail
= 1 - 0.9995*0.9997*0.9998
= 0.00099969
Therefore 0.00099969 is the required probability here.
b) Probability that the rocket will successfully fire the next 100 times that it is required is computed as:
= Probability that it will be a successful firing * Probability that it will be a successful firing *Probability that it will be a successful firing *.................... 100 times
= ( 1 - 0.00099969)100
= 0.9048
Therefore 0.9048 is the required probability here.
c) Here we are given that the failure of the components ( three of them ) are disjoint events. Therefore they cannot happen together. Therefore the required probability here is computed as:
= Probability that component 1 fails + Probability that component 2 does fails + Probability that component 3 does fails
= (1 - 0.9995)+ (1-0.9997)+ (1-0.9998)
= 0.0005+ 0.0003 + 0.0002
= 0.001
Therefore the required probability of failure here is 0.001