Question 192
A calculator has a display, which shows a nonnegative integer $N$, and a button, which replaces $N$ by a random integer chosen uniformly from the set $\{0,1, \ldots, N-1\}$, provided that $N\gt 0$. Initially, the display holds the number $N=2003$. If the button is pressed repeatedly until $N=0$, what is the probability that the numbers $1,10,100$, and $1000$ will each show up on the display at some point?