Nice statistical problem! I only thought of a solution that is valid for many cars and few worksheet spots. Were you able to fully solve it on the spot?
The best solution I could think of on the spot was this:
First write down every license place until it fills up. Then for each time a car drives by, give it a 1/2 chance to write it down. Moving down the list regardless of the outcome. Then once the bottom is reached, do it again but with 1/3 a chance. Over and over again, decreasing the chance each time.
I checked for which situations it is valid, and it surprisingly was when there are many more cars than the list size. But it is not valid when there are just a few more cars than the list size.
Mine was to roll a number between 1 and N once N is greater than M (N being cars seen so far, M your slots available). If you get a number that is less than or equal to M, replace that slot. If it is greater than M, that sample is dropped.
