To protect account access, we allot to every user of the system private 4-digit number (PIN). At this, we select each of four digits independently and uniform at random from {0,...,9}. Assume that we have 30 users U1, ..., U30 of this system.
i) Find probability that at least two of users have identical PINs?
ii) In reply to concerns about "small PIN values", we change PIN generation procedure as follows: Whenever the uniform random choice for first digit results in a 0, we use a 1 instead. With the new PIN generation, find probability of user U5 having PIN 1301?