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Every man inherits his haplo-type from his father, and thus the male lineage can be traced back.

Given a population $n$ males,

At each generation we uniformly sample $n$ males (with repetitions) to be the fathers of the next male generation.

After how many generations will the population converge to only one haplo-group ?

Uri Goren
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    Nobody can answer this question, it is way too general. Do you have a context? If you were to model the number of male sons of an individual by a Galton-Watson process, your question boils down to "what's the probability to have exactly one tree that survives". Is it what you are thinking? – Graffitics Feb 22 '16 at 18:57
  • Assume that there are $n$ sons in each generation, and you are choosing a father for each son from the previous generation – Uri Goren Feb 22 '16 at 19:00

2 Answers2

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Not the complete answer but something to help you get started.

Let's write the probability to go from $k$ different haplo-types to $l<k$ haplo-types in one step, when there are $n$ initial haplo-types. We get the probability (the reasoning can be found in this answer): $$\Pr[k\to l;n] = \frac{n!S_{n,l}}{(k-l)!\times k^n}$$ with $S_{n,l}$ denoting the Stirling number of the second kind.

To get the probability expected number of surviving haplo-types at the beginning of the second step, you can use the formula $\sum_{i=1}^{n}i \times Pr[n \to i;n]$.

Using these probabilities you can describe your problem as a Markov Chain, from which you can estimate the expected time it takes to eliminate all but one haplo-type.

Community
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Graffitics
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  • The expected number of haplo-types at the second generation is $$n(1-(1-\frac{1}{n})^n)$$, how can this can be seen from the formula you suggested ? – Uri Goren Feb 23 '16 at 21:27
  • @UriGoren Please disregard the abomination I had written before. Now this is correct, but still a long way from the answer. At the same time, you could give it better tags/description, as you just have an urn problem, and nothing to do with biology. – Graffitics Feb 24 '16 at 19:52
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This is the Wright-Fisher model. You can find the expected time for one type to take over the population on Wikipedia, or, if you want something more official, Ewens' Mathematical population genetics, eqn. (3.11). It's $E[T]=-2n^2(1-1/n)\log(1-1/n)$, which is $E[T]\approx 2n$ for $n\gg 1$.

Daniel Weissman
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