= 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 is
$$
p_{G+\mathrm{evolved}}=\int_0^1(g^2+2gw)dy=19\int_0^1\frac{y^2}{8100}dy=\frac{19}{24300}.
$$
This 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>, is
$$
\boxed{\frac{p_{G+\mathrm{evolved}}}{p_{\geq1G}}=\frac{19}{269}\simeq0.07063.}
$$
In particular the denominator is a fraction of systems, not the individual-star fraction $1/180$.
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