Yes, Laplace's rule works in this instance. Assume you have a printer that prints out papers that say either "yes" or "no", each independently identically distributed with unknown (and uniformly distributed) p. If you pull the papers directly from the printer you have a classic Laplace's rule situation. If you print out N papers without looking at them, then look at each in turn, the situation is essentially unchanged. Furthermore, the probability that k of the N papers say "yes" is the same for each 0 <= k <= N.
This is a question really, not a post, I just can't find the answer formally. Does laplace's rule of succession work when you are taking from a finite population without replacement? If I know that some papers in a hat have "yes" on them, and I know that the rest don't, and that there is a finite amount of papers, and every time I take a paper out I burn it, but I have no clue how many papers are in the hat, should I still use laplace's rule to figure out how much to expect the next paper to have a "yes" on it? or is there some adjustment you make, since every time I see a yes paper the odds of yes papers:~yes papers in the hat goes down.