In a support system in the U.S. space program, a single crucial component works only 85% of the time. In order to enhance the reliability of the system, it is decided that 3 components will be installed in parallel such that the system fails only if they all fail. Assume the components act independently and that they are equivalent in the sense that all 3 of them have an 85% success rate. Consider the random variable X as the number of components out of 3 that fail.
(a) Write out a probability function for the random variable X.
(b) What is E(X) (i.e., the mean number of components out of 3 that fail)?
(c) What is Var(X)?
(d) What is the probability that the entire system is successful?
(e) What is the probability that the system fails?
(f) If the desire is to have the system be successful with probability 0.99, are three components sufficient? If not, how many are required?