Question: A hacker is trying to break into a password-protected website by randomly trying to guess the password. Let m be the number of possible passwords.
(a) Suppose for this part that the hacker makes random guesses (with equal probability), with replacement. Find the average number of guesses it will take until the hacker guesses the correct password (including the successful guess).
(b) Now suppose that the hacker guesses randomly, without replacement. Find the average number of guesses it will take until the hacker guesses the correct password (including the successful guess).
(c) Show that the answer to (a) is greater than the answer to (b) (except in the degenerate case m = 1), and explain why this makes sense intuitively.
(d) Now suppose that the website locks out any user after n incorrect password attempts, so the hacker can guess at most n times. Find the PMF of the number of guesses that the hacker makes, both for the case of sampling with replacement and for the case of sampling without replacement.