Where do $\frac{1-r}{2}$ and $\frac{r}{2}$ come from?

Consider the coupling double heterozygote, $A_1B_1/A_2B_2$. When recombination doesn't happen, $A_1B_1$ and $A_2B_2$ occur in equal frequency ($1/2$), and $A_1B_2$ and $A_2B_1$ don't occur at all. When recombination happens, the four possible gametes occur in equal frequency ($1/4$). So the recombination frequency,⁴ $r$, is half the crossover frequency,⁵ $c$, i.e., $r = c/2$. Now the results of crossing over can be expressed in this table:

Frequency	$A_1B_1$	$A_1B_2$	$A_2B_1$	$A_2B_2$
$1-c$	$\frac{1}{2}$	0	0	$\frac{1}{2}$
$c$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$	$\frac{1}{4}$
Total	$\frac{2-c}{4}$	$\frac{c}{4}$	$\frac{c}{4}$	$\frac{2-c}{4}$
	$\frac{1-r}{2}$	$\frac{r}{2}$	$\frac{r}{2}$	$\frac{1-r}{2}$