probability of getting same person

How would I calculate the probability of randomly selecting a house and getting the current owner’s last naming same as previous owner’s last name? For example, let’s say 1% of the population has the last name “Doe” and picking a house randomly getting the current owner’s name is Doe and the previous owner’s name was Doe too. Thanks

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Not sure how is it different from you previous question: math.stackexchange.com/q/270455/45813 – jay-sun Jan 4 '13 at 22:21
If you don't restrict to the case that Doe moves out, that is exactly what I answered to your other question. It is the sum of the squares of the probabilities of each name in the population. – Ross Millikan Jan 4 '13 at 22:52
Please just edit your one question to make it as you desire instead of creating multiple, separate variations of the same question. – JohnD Jan 7 '13 at 19:57

Assume that in the general population, the names are $N_1,N_2,\dots, N_n$, and they occur with probabilities $p_1,p_2,\dots,p_n$.
Then, under certain assumptions, the required probability is $$p_1^2+p_2^2+\cdots+p_n^2.$$ To complete the calculation, you would need to know the $p_k$.