  We propose a Markov chain approach for the evolution of a genealogical line
of genomes. Our idealized genome has $N$ sites and each site can be in state
$0$ or $1$. At each time step we pick a site at random. If the site is in state
$0$ we flip it to state 1 with probability $p$ or we keep it in state $0$ with
probability $1-p$. If the site is in state $1$ we flip it to state 0 with
probability $1-p$ or we keep it in state $1$ with probability $p$. Even when
state 1 has a selective advantage (i.e. $p>1/2$) the Markov chain is quite
unlikely to approach the most fit allele (i.e. all 1's). In fact, randomness
(i.e. which site is picked for a possible mutation) and selection (i.e. the
value of $p$) balance each other out so that the number of $1$'s in the genome
converges to a Gaussian distribution centered around $Np$.
