Answer to Question #261965 in Statistics and Probability for tiara

Question

A satellite can fail for many possible reasons, two of which are computer failure and engine failure. F or a given mission, it is known that: The probability of engine failure is 0.008. The probability of computer failure is 0.001. G iven engine failure, the probability of satellite failure is 0.98. G iven computer failure, the probability of satellite failure is 0.45. G iven any other component failure, the probability of satellite failure is zero. (a) D etermine the probability that a satellite fails. (b) D etermine the probability that a satellite fails and is due to engine failure. (c) Assume that engines in different satellites perform independently. G iven a satellite has failed as a result of engine failure, what is the probability that the same will happen to another satellite?

Answer 1

Define the following events,

Let,

$C$ be the event of a computer failure

$E$ be the event of an engine failure

$S$ be the event of a system failure

We have the following probabilities,

p(E)=0.008

p(C)=0.001

and the conditional probablities are given as,

p(S|E)=0.98

p(S|C)=0.45

a.

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

$p(S)=p(S|E)*p(E)+p(S|C)*p(C)$

$p(S)=(0.008*0.98)+(0.001*0.45)=0.00829$

Probability that the system fails is 0.00829

b.

 Probability that a satellite fails and is due to engine failure is given as $p(E\cap S)$and can be found using the formula below

$p(E\cap S)=p(S|E)*p(E)=0.008*0.98=0.00784$

Therefore, the probability that the system fails and is due to engine failure is 0.00784

c.

If engines in different satellites work independently then $p(E\cap S)$ will be equal for each satellite therefore,

$p(E\cap S)=p(E| S)*p(S)=0.008*0.98=0,00784$

Given a satellite has failed as a result of engine failure, the probability that the same will happen to another satellite is 0.00784.