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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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 $p_k$.
This has already been pointed out. What I would like to mention is that the calculation is unlikely to give an answer that is anywhere near the truth.
Many transfers of property are among close relatives, including transfers from parents to children. Many other transfers, in small towns, are between residents of these towns, and the variety of names in small communities is much smaller than the overall variety. Also, there is a significant degree of de facto ethnic segregation, again reducing the effective variety of names in property transfers.
So I would expect that the actual proportion of same name transfers is significantly higher than the one that would be predicted by the formula above.