fixation event.

Finally, we observed that the probability for any beneficial mutant to go to fixation is equal to the probability that a genotype bearing [*all*]{} mutations will become the ancestor of the entire population. Moreover, this observation holds for an arbitrary number of competing mutations, even in the presence of deleterious mutations, independently of the order of appearance of the beneficial mutations. Naturally, while such a probability is very high if the sum of all benefits is large, the probability of a single sequence appearing with this combination of mutations is exponentially small in non-recombining populations. This result was confirmed using an extension of Kimura’s result [-@kimura62] to finite mutation rates \[Eq. (\[kimura-approx\])\] and also by means of a branching process formulation. Both approaches yield similar results and are in excellent agreement with our numerical simulations. We find that the branching process approach is more accurate at low mutation rates, away from the error threshold. As it is formulated in the $N \rightarrow \infty$ limit, it fails to predict the asymptotic result $\pi
\approx n/N$ which is reached in the limit that all $\gamma_{i}$ vanish.

The present analysis concerns the rate of adaptation as beneficial mutations compete against each 