Coeval binary population

A coeval binary has components formed at the same time. A population of such binary stars can still have a distribution of system ages. If component masses have independence and the state at age $t$ is determined separately for each component, the states have conditional independence given $t$. If $g(t)$ and $w(t)$ are the conditional probabilities of two states, the unconditional unordered pair fractions are $\mathbb{E}[g(t{)}^2]$ and $2\mathbb{E}[g(t)w(t)]$, not generally products of marginal state fractions.

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.

In an instantaneous burst all binary components have a fixed common age. If the initial component masses are independent, the probability of two red giants is the square of the individual giant probability at that age. This need not equal the age-averaged pair fraction in a continuously forming population.

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.