This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.

The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_1}$ and at the limit will reach $(1-z)$.

If we denote $p_k$ as the probability of someone in generation $k$ to be related to the initial population. And for simplicity lets relax the siblings & cousins rule (can be added later). Then: $$p_1 = {2 \over n_1}$$

As each person in the new generation has exactly 2 ancestors in the initial population. $$ p_2 = relatives \times {2 \over n_2} + non.relatives \times {4 \over n_2} $$ In this case relatives could be calculated as: $$ relatives = {\binom{c}{2} \times {n \over c} \over \binom{n}{2}} = {c-1 \over n-1}$$ Or in other words, the number of sibling combinations, times the number of siblings family, divided by the total mating combinations. $$p_3 = immediate.relatives \times {4 \over n_3} + cousins \times {6 \over n_3} + non.relatives \times {8 \over n_3}$$

With each generation, the probability to be related to someone at the initial population will undoubtedly grow, but at a decreasing pace. This is because the probability to draw "relatives" which are coming from the same or similar tree will grow.

Lets use ethnicity as an example. Lets say we know for a fact someone is 100% Caucasian. At generation 28 he is most likely related to a significant portion of the Caucasian population in 1300 (As shown by @whuber simulation). Lets say he is marrying someone who is 100% of a different ethnicity. Their offspring will be linked to approximately double the number of people they are linked to from 1300.

Another interesting thought is that given the human (homosapien) race started from ~600 people in Africa, then we are most likely a genetic permutation of all of them who successfully mated.

This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.

The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_1}$ and at the limit generation will approach $\lim_{k \to \infty } p_k = (1-z)$.

If we denote $p_k$ as the probability of someone in generation $k$ to be related to the initial population. And for simplicity lets relax the siblings & cousins rule (can be added later). Then: $$p_1 = {2 \over n_1}$$

As each person in the new generation has exactly 2 ancestors in the initial population. $$ p_2 = relatives \times {2 \over n_2} + non.relatives \times {4 \over n_2} $$ In this case relatives could be calculated as: $$ relatives = {\binom{c}{2} \times {n \over c} \over \binom{n}{2}} = {c-1 \over n-1}$$ Or in other words, the number of sibling combinations, times the number of siblings family, divided by the total mating combinations. $$p_3 = immediate.relatives \times {4 \over n_3} + cousins \times {6 \over n_3} + non.relatives \times {8 \over n_3}$$

With each generation, the probability to be related to someone at the initial population will undoubtedly grow, but at a decreasing pace. This is because the probability to draw "relatives" which are coming from the same or similar tree will grow.

Lets use ethnicity as an example. Lets say we know for a fact someone is 100% Caucasian. At generation 28 he is most likely related to a significant portion of the Caucasian population in 1300 (As shown by @whuber simulation). Lets say he is marrying someone who is 100% of a different ethnicity. Their offspring will be linked to approximately double the number of people they are linked to from 1300.

Another interesting thought is that given the human (homosapien) race started from ~600 people in Africa, then we are most likely a genetic permutation of all of them who successfully mated.

This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.

The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_1}$ and at the limit will reach $(1-z)$.

If we denote $p_k$ as the probability of someone in generation $k$ to be related to the initial population. And for simplicity lets relax the siblings & cousins rule (can be added later). Then: $$p_1 = {2 \over n_1}$$

As each person in the new generation has exactly 2 ancestors in the initial population. $$ p_2 = relatives \times {2 \over n_2} + non.relatives \times {4 \over n_2} $$ In this case relatives could be calculated as: $$ relatives = {\binom{c}{2} \times {n \over c} \over \binom{n}{2}} = {c-1 \over n-1}$$ Or in other words, the number of sibling combinations, times the number of siblings family, divided by the total mating combinations. $$p_3 = immediate.relatives \times {4 \over n_3} + cousins \times {6 \over n_3} + non.relatives \times {8 \over n_3}$$

With each generation, the probability to be related to someone at the initial population will undoubtedly grow, but at a decreasing pace. This is because the probability to draw "relatives" which are coming from the same or similar tree will grow.

Lets use ethnicity as an example. Lets say we know for a fact someone is 100% Caucasian. At generation 28 he is most likely related to a significant portion of the Caucasian population in 1300 (As shown by @whuber simulation). Lets say he is marrying someone who is 100% of a different ethnicity. Their offspring will approximately double the number of people they linked to from 1300.

Another interesting thought is that given the human (homosapien) race started from ~600 people in Africa, then we are most likely a genetic permutation of all of them who successfully mated.

This is a very interesting question as it is asking us to mathematically solve a fractal. Such as the famous game of life.

The % of the population which each generation related to will grow over each iteration, starting at $p_1={2 \over n_1}$ and at the limit will reach $(1-z)$.

If we denote $p_k$ as the probability of someone in generation $k$ to be related to the initial population. And for simplicity lets relax the siblings & cousins rule (can be added later). Then: $$p_1 = {2 \over n_1}$$

As each person in the new generation has exactly 2 ancestors in the initial population. $$ p_2 = relatives \times {2 \over n_2} + non.relatives \times {4 \over n_2} $$ In this case relatives could be calculated as: $$ relatives = {\binom{c}{2} \times {n \over c} \over \binom{n}{2}} = {c-1 \over n-1}$$ Or in other words, the number of sibling combinations, times the number of siblings family, divided by the total mating combinations. $$p_3 = immediate.relatives \times {4 \over n_3} + cousins \times {6 \over n_3} + non.relatives \times {8 \over n_3}$$

With each generation, the probability to be related to someone at the initial population will undoubtedly grow, but at a decreasing pace. This is because the probability to draw "relatives" which are coming from the same or similar tree will grow.

Lets use ethnicity as an example. Lets say we know for a fact someone is 100% Caucasian. At generation 28 he is most likely related to a significant portion of the Caucasian population in 1300 (As shown by @whuber simulation). Lets say he is marrying someone who is 100% of a different ethnicity. Their offspring will be linked to approximately double the number of people they are linked to from 1300.

Another interesting thought is that given the human (homosapien) race started from ~600 people in Africa, then we are most likely a genetic permutation of all of them who successfully mated.