I have a nation of population 60 million sharing a finite set of first names ($F$) and last names ($L$). Let's say I have a sample of 3 million people from that nation for which I know first names and last names. From this information I can build a dictionary of first and last names which are a reasonable approximation of $F$ and $L$ and assign a probability to each name and surname (assuming name and surname are independent). So if in my sample I have 2000 Peters and 3000 Griffins, I assign to a random person $X$ probabilities $P$($X$ is Peter) = $2000/3000000$ = $0.00067$ and $P$($X$ is Griffin) = $3000/3000000$ = $0.001$.
My question is, if I have another sample of size $n$ from the same population, how do I calculate the probability that I will encounter in the sample at least two people named Peter Griffin?