A hospital administrator worries about the potential loss of electric power as a result of a power blackout. The hospital, of course, has a standby generator, but it too is subject to failure, having a mean time between failures of 500 hours. It is reasonable to assume that the time between failures is exponentially distributed.
a. What is the probability that the standby generator fails during the next 24-hour blackout?
b. Suppose the hospital owns two standby generators that work independently of one another. What is the probability that both generators fail during the next 24-hour blackout?
Mean = $\frac{500}1 = 500$ Rate parameter = $\frac{1}{500} = 0.002$
I understand part a:
a. $P(X<24)= 1-e^{(-0.002*24)}= 1-0.9531= 0.0469$
My question is about part b:
Would this go back to conditional probability e.g. P(A|B)?
My nephew confirmed conditional probability here. Since both generators have the same rate of failure, P(A|B) = pq where p=Pa(X<24) and q=Pb(X<24) = 0.0469 * 0.0469 = 0.0022