Engineering system of type k-out-of-n is operational if at least k out of n components are operational. Otherwise, the system fails. Suppose that a k-out-of-n system consists of n identical and independent elements for which the lifetime has Weibull distribution with parameters r and λ. More precisely, if T is a lifetime of a component, P(T ≥ t) = e−λtr, t ≥ 0. Time t is in units of months, and consequently, rate parameter λ is in units (month)−1. Parameter r is dimensionless.
Assume that n = 8,k = 4, r = 3/2 and λ = 1/10.
(a) Find the probability that a k-out-of-n system is still operational when checked at time t = 3.
(b) At the check up at time t = 3 the system was found operational. What is the probability that at that time exactly 5 components were operational?