Answer to Question #265838 in Statistics and Probability for Muhammad Ali

Question

1.A satellite can fail for many possible reason, two of which are computer failure and engine failure. For a given mission, it is known that:

 The probability of engine failure is 0.008.

 The probability of computer failure is 0.001.

 Given engine failure, the probability of satellite failure is 0.98.

 Given computer failure, the probability of satellite failure is 0.45.

 Given any other component failure, the probability of satellite failure  is zero.

(a)Determine the probability that a satellite fails.   (Soo.2.11)

(b)Determine the probability that a satellite fails and is due to engine failure.

(c)Assume that engines in different satellites perform independently. Given a satellite has failed as a result of engine failure, what is the probability that the same will happen to another satellite?

Answer 1

Let $A$ be "computer failure"

$B$ be "engine failure"

$C$ be "system failure"

(a) The probability that a satellite fails can be found using the law of total probability.

$P(C)=P(C|A)*P(A)+P(C|B)*P(B)=0.001*0.45+0.008*0.98=0.00829$

Probability that the system fails is 0.00829

(b) Probability that a satellite fails due to engine failure could bo found using bayesian formula

$P(B|C)={\frac {P(C|B)*P(B)} {P(C)}}={\frac {0.98*0.08} {0.0829}}=0.9457$

(c) Let D be "computer will fall due to engine failure". Since engines work independently, then the probabilities for given sattelite doesn't change whatever happens to another one, then

$P(D) = P(B)*P(C|B)=0.008*0.98=0.00784$