In an ideal population
with effective size N_{e}, the inbreeding
coefficient is simply

F
= 1 / 2N_{e}

This population then comprises 2N_{e
}alleles at any locus: formation
of diploid individuals is a random draw &
replacement exercise from this gene
pool. For any individual, the probability that the first allele
drawn is the same as itself is 1 [think about it]: the probability of drawing the same
allele again is simply the reciprocal of the gene pool size,
thus 1 / 2N_{e}.

In a finite population with effective size N_{e}AND some degree of inbreeding, the inbreeding
coefficientF at time t (F_{t})
is related to that in the previous generation F_{t-1}
by a recursion equation,

F_{t}
= (1/2N_{e})(1) + (1 - 1/2N_{e})(F_{t-1})
.
The first term applies to inbred individuals,
the second to non-inbred individuals, as follows.
[Notice that this term is the same as the first equation,
above]. The expectation of drawing any particular allele once
is 1 / 2Ne. If that allele occurs in an inbred individual,
then the expectation that the second allele is identical by
descent with the first is necessarily 1.
Otherwise, the expectation that the second allele chosen is not
the same asthe first is (1 - 1/2N_{e}).
In this case, the probability that the second allele drawn would
be identical with the first would by definition be zero,
except that some fraction of the population was
inbred in the previous generation F_{t-1}, so the probability of identity by
descent is modified. The recursion equation then continues to
similar terms for t-2, t-3, and so on. Considering the whole series, the inbreeding
coefficient at equilibrium F_{e}
is F_{t}
= F_{t-1} = F_{e} .
[The form of the equation
is identical with that previously used to estimate the
expectations of f(AA) & f(BB)
under inbreeding].

Now consider how mutation changes
the probability of identity by descent. The expectation that
any single allele mutates is µ.
Among individuals with alleles identical by descent, the
expectation that neither allele mutates (and thus
remain identical by descent with each other) is (1 - µ)^{2}.
At equilibrium F_{e},

This is an extremely important
equation in evolutionary population genetics, because it
enables estimation of population size from observed
genetic data, or vice versa.