A computer system has n users, each with a unique name and password. Due to a software error, the n passwords are randomly permuted internally (i.e., each of the n! possible permutations is equally likely). Only those users lucky enough to have had their passwords unchanged in the permutation are able to continue using the system.
(a) What is the probability that a particular user, say user 1, is able to continue using the system?
(b) What is the expected number of users able to continue using the system? Hint: Let Xi be a rv with the value 1 if user i can use the system and 0 otherwise.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.