Consider humans (or really any creature with sufficiently complex set of genes). If person $A\_0$ has an offspring, say $A\_1$, we can without a doubt find markers in $A\_1$ that we can definitely say indicate that $A\_1$ is the progeny of $A\_0$. Likewise, for any generation $n$ we can say that $A\_{n+1}$ has markers that indicate that generation $A\_n$ is their parent. Presumably (and maybe this is where I'm heavily mistaken), as $n$ grows large there are fewer "direct signs" that $A\_n$ has $A\_0$ in their ancestry. I mention "direct signs" (although this isn't an official term) because obviously if one can prove $A\_{n-1}$ is the parent generation to $A\_n$ and $A\_{n-2}$ proceeds $A\_{n-1}$ and so on ... then one can deduce the relation between $A\_0$ and $A\_n$. That's not my point. My question is this: Do the genes of $A\_0$ "dilute" so that after so many generations it is possible or probable that none of the genes from $A\_0$ are present in $A\_n$ for some large $n$ (where by "none" I mean "no genes that give us a high warrant for believing $A\_n$ is related to $A\_0$ in the absence of evidence about $A\_j$ for $2\le j \le n-1$"). Obviously if $A\_0$ is human and $A\_n$ is human then they share a great deal of their genes with one another (something like $99.99\%$)

If there is such an $n$, how large does $n$ need to be so that say $90\%$ of what makes $A\_0$ unique is no longer present in $A\_n$ (with a high probability of confidence), what about $95\%$ no longer being present, or $99\%?$

Please let me know if my question isn't sufficiently clear (I'm trying to convey the heart of my question, even if it isn't perfectly worded).