  The formula for the probability of fixation of a new mutation is widely used
in theoretical population genetics and molecular evolution. Here we derive a
series of identities, inequalities and approximations for the exact probability
of fixation of a new mutation under the Moran process (equivalent results hold
for the approximate probability of fixation for the Wright-Fisher process after
an appropriate change of variables). We show that the behavior of the logarithm
of the probability of fixation is particularly simple when the selection
coefficient is measured as a difference of Malthusian fitnesses, and we exploit
this simplicity to derive several inequalities and approximations. We also
present a comprehensive comparison of both existing and new approximations for
the probability of fixation, highlighting in particular approximations that
result in a reversible Markov chain when used to model the dynamics of
evolution under weak mutation.
