Empirical probability question: if you have N children what is the expected length L of the longest chain of descendants you might have? Ie how long before your lineage dies out? Make suitable assumptions about distributions of offspring. Feels like a 101 question.
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I think the keyword you want is Galton-Watson process: https://en.wikipedia.org/wiki/Galton%E2%80%93Watson_process …
This caused me to look up my surname which has apparently been going strong for 2700 years if you believe Chinese genealogies 
https://en.wikipedia.org/wiki/Yuan_(surname) …
