Question: Many vehicles used in space travel are constructed with redundant systems to protect flight crews and their valuable equipment. In other words, backup systems are included within many vehicle components so that if one or more systems fail, backup systems will assure the safe operation of the given component and thus the entire vehicle. For example, consider one particular component of the U.S. space shuttle that has n duplicated systems (i.e., one original system and n -1 backup systems). Each of these systems functions, independently of the others, with probability 0.98. This shuttle component functions successfully provided that at least one of the n systems functions properly.
a. Find the probability that this shuttle component functions successfully if n = 2.
b. Find the probability that this shuttle component functions successfully if n = 4.
c. What is the minimum number n of duplicated systems that must be incorporated into this shuttle component to ensure at least a 0.9999 probability of successful operation?