Conditional evolved-companion fraction in a coeval binary population ($\mathbb{E}[g^2+2gw]/\mathbb{E}[2g-g^2]$)

If $g(t)$ is the conditional giant fraction and $w(t)$ a second evolved-state fraction at common age $t$, independent component masses give the fraction of giant-containing systems with another evolved component as $\mathbb{E}[g^2+2gw]/\mathbb{E}[2g-g^2]$. Averaging must follow conditioning on age, rather than multiplying marginal state probabilities.