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.