Continuous-birth binary state fractions ($P(GG):P(GW):P(WW)=a^2:2ab:b^2$)

Suppose binary ages have uniform coordinate $y\in [0,1]$ and independently sampled component masses give disjoint conditional state fractions $g(y)=ay$ and $w(y)=by$. Integrating the conditional pair probabilities gives $P(GG)=a^2/3$, $P(GW)=2ab/3$, and $P(WW)=b^2/3$. The fraction with at least one $G$ is $a-a^2/3$, and the fraction of those systems with a second evolved component is $(a^2+2ab)/(3a-a^2)$. Components share their age, so their unconditional states are correlated even when their birth masses have independence.