 Solution

Use conditional independence at the common age $Y=y$. A system with a red giant and a second evolved component is either a pair of red giants, with conditional probability $g(y{)}^2$, or a red giant and a white dwarf, with conditional probability $2g(y)w(y)$. The factor two counts which component is the white dwarf; a pair of red giants is counted only once. Since $w=9g$, the unconditional probability of this event isThis event is contained in the event of at least one red giant. Therefore the requested conditional probability, among systems selected for containing a red giant, isIn particular the denominator is a fraction of systems, not the individual-star fraction $1/180$.