Haldane showed many years ago that the probability of fixation of new advantageous mutation is approximately 2s, if s is small, where s is the strength of selection in favour of the new mutation. It has never been very clear to me, until recently, why the probabiility of fixation is independent of the effective population size, Ne. I think I have now worked this out. The critical phase for the survival of an advantageous mutation is when it's rare, and when it's rare, its dynamics are well approximated by a Poisson distribution. As such the variance in the number of copies of the allele from generation to generation, due to drift, depends upon the absolute number of copies of the allele in the population, it does not depend upon the frequency of the allele or the size of the population.

Kimura generalised Haldane's result and showed that the probability of fixation of new advantageous mutation is actually approximately 2Nes/N, where N is the census population size (Haldane's results only applies when Ne = N). This makes sense under the explanation above, because only a proportion Ne/N individuals contribute to the next generation; so in considering the variance in the number of copies of the allele we should only consider those from the proportion of individuals contributing to the next generation.