description of the exact time dependence (see Numerical Results) without free parameters.

Overall probability of fixation
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Besides the individual probabilities $\pi_1(\Delta t)$ and $\pi_2(\Delta t)$, their sum $\pi=\pi_1(\Delta t)+\pi_2(\Delta t)$ is also of interest. This sum is the overall probability that at least one mutant goes to fixation. When we sum Eqs. (\[p1minusinf\]) and (\[p2minusinf\]), or Eqs. (\[p1inf\]) and (\[p2inf\]), we find that the overall fixation probability does not depend on the order in which the mutants are introduced, and has the form $$\label{overall-fix-prob}
  \pi = P(s_1,u)+P(s_2,u)-P(s_1,u)P(s_2,u)\,.$$ Furthermore, if Eq. (\[tanh-ansatz\]) is indeed a good description of $\pi_i(\Delta t)$ for arbitrary $\Delta t$, then $\pi$ should not depend on $\Delta t$ at all, because all time dependencies cancel when we use Eq. (\[tanh-ansatz\]) to calculate $\pi_1(\Delta t) + \pi_2(\Delta t)$.

The invariance of the overall fixation probability $\pi$ under the order by which the mutants appear is a general property. It holds also when more than 2 mutants arise: Assume that $n$ advantageous mutants arise, with selective advantages $s_1,\dots,s_n$ compared to the wild type. Further assume that the time intervals between the appearances of the mutants are large. Then, the probability that none of the mutants make it to 