Hi please use the given information to answer each part of the problem and please include a short explanation, drawing, or steps/formulas to improve my understanding of the problem. Thank you!

Consider a hierarchical storage . _ system composed of n subsystems, each comprlsmg k servers. Each subsystem can tolerate a single server failure, and the overall system can tolerate a single subsystem failure. Thus, in order for the overall system to fail, there has to be at least two sub- systems that each have at least two server failures. Suppose servers fail independently with probability p. (a) Let X be the number of servers that failed in a particular subsystem. Find the pmf of X in terms of k and p. What is the probability p0 that a particular subsystem fails? (b) Let Y be the number of subsystems that failed. Find the pmf of Y in terms of n and p0. What is the probability p1 that the overall system fails? (0) Give upper bounds on p0 and p1 using the union of events bound. Hint: Let Aij be the event both server 2' and j fail for 1 g i