evolution - Hardy-Weinberg equilibrium generalized to add inbreeding (non-random mating)

Background

Quoting from Gillespie's book

The level of inbreeding is measured by the inbreeding coefficient, $F_I$, which is the probability that two alleles in an individual are identity by descent.

The two possible alleles are $A_1$ and $A_2$ and their frequencies are $p$ and $q=1-p$, respectively. Gillespie goes on and give the expected genotype frequencies given $p$ (and $q$) and $F_I$

$$ \begin{array}{r|c|c|c} Genotype & A_1A_1 & A_1A_2 & A_2A_2 \\ \hline Frequency & p^2(1-F_I)+pF_I & 2pq(1-F_I) & q^2(1-F_I)+qF_I \\ \end{array} $$

Question

Can you please help me to understand why those are the appropriate genotype frequencies?

Example of what is unclear to me

By the definition of $F_I$, I was expecting that the frequency of `$A_1A_2$ (and $A_2A_1$) would be $(1-F_I)$, that is the frequency (or probability) that two alleles in an individual are not identical by descent. I feel like the definition given above is wrong. Should it rather be something like the weight for sampling like individuals? That is if there were only two possible mates, one that is $A_1A_1$ and one that is either $A_1A_2$ or $A_2A_2$, then the probability for a $A_1A_1$ individual to mate with the other $A_1A_1$ is $F_I$.

Answer

You're confused because you're failing to distinguish between 'identical' and 'identical by descent'. Some pairs of alleles would still be identical even in the absence of inbreeding.

We model the inbreeding by classifying allele pairs as IDB - always homozygous - or not IDB - distributed according to Hardy Weinberg. The frequency of a pair of alleles being A₁,A₂ (or A₂,A₁) is thus the frequency under H.W.E. (2pq) multiplied by the chance of them NOT being IBD (1-F_I). And conversely, a pair can be homozygous and not IBD - at frequency p ²(1-F_I), or because they are IBD - at frequency p F _I.