I have $N$ bins labeled "1" through "N". I place $k$ balls in bin "1", then successively move balls from bin "1" to bin "2", bin "2" to bin "3", and so on to bin "N". However, while moving any particular ball from one bin to another, I discard the ball with probability $(1 - p)$.
What is the probability distribution for the number of balls in bin "N"?