= Solution
A constant number <star formation rate> over $10\,\mathrm{Gyr}$ produces a uniform present-age distribution. The <birth-age distribution under constant star formation> therefore gives
$$
\boxed{Y(t)=\frac{t}{10\,\mathrm{Gyr}}=0.1\frac{t}{\mathrm{Gyr}}\quad(0\le t\le10\,\mathrm{Gyr}).}
$$
Outside this age interval the <cumulative distribution function> is zero or one. The census here retains stellar remnants as objects, as required by the population model.
Write $m=M/M_\odot$. The <initial mass function> is proportional to $m^{-3}$ on $m\ge0.2$. Normalizing the tail integral gives the <power-law initial-mass-function tail coordinate>:
$$
\boxed{X(m)=\frac{\int_m^\infty s^{-3}ds}{\int_{0.2}^\infty s^{-3}ds}
=\frac{0.04}{m^2}\quad(m\ge0.2).}
$$
For lower thresholds $X=1$. Because the birth <mass> and formation time have <independence>, the <probability integral transform> makes $(X,Y)$ uniform on the unit square. $X$ is an upper-tail coordinate, so larger masses have smaller $X$.
A <red giant> of current age $t$ lies in the <stellar lifetime regions in a mass-age diagram>
$$
\frac{8}{m^2}\le\frac{t}{\mathrm{Gyr}}<\frac{10}{m^2},\qquad 0\le\frac{t}{\mathrm{Gyr}}\le10.
$$
There are no <red giants> below $m=\sqrt{0.8}$ in this model. For $\sqrt{0.8}<m<1$ the upper boundary is the population's age limit, while for $m\ge1$ both lifetime boundaries are present. A <white dwarf> lies above $t/\mathrm{Gyr}=10/m^2$.
In uniform <probability> coordinates the main-sequence boundary is $Y=20X$ and the end of the giant phase is $Y=25X$. The giant region is therefore $20X\le Y<25X$, clipped to the unit square: it is the triangle with vertices $(0,0),(1/25,1),(1/20,1)$. Its area, and hence the individual giant fraction, is
$$
\boxed{\Pr(G)=\frac12\left(\frac1{20}-\frac1{25}\right)=\frac1{200}=0.5\%.}
$$
The <white dwarf> region $Y\ge25X$ is the triangle with vertices $(0,0),(0,1),(1/25,1)$, giving
$$
\boxed{\Pr(W)=\frac12\frac1{25}=\frac1{50}=2\%.}
$$
The remaining $97.5\%$ are on the <main sequence>. Boundaries have zero <probability> and do not affect these counts. Treating every post-giant object as a <white dwarf> is a stipulated toy-model assumption; real high-mass stars have other <stellar remnants>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-42-population-regions.png]
{title=Red-giant and white-dwarf regions in mass-age coordinates and their triangular images in uniform tail-mass and age coordinates}
For the binary calculations, use a <coeval binary population>: components share their birth time even though their birth masses are independently drawn and their subsequent evolution is independent. Hence $(X_1,X_2,Y)$ is uniform in the unit cube. At a fixed age coordinate $Y=y$, a component is a giant when $y/25<X<y/20$, and a <white dwarf> when $0<X<y/25$. Define the conditional state fractions
$$
g(y)=\frac{y}{100},\qquad w(y)=\frac{y}{25},\qquad e(y)=g(y)+w(y)=\frac{y}{20}.
$$
Components have <conditional independence> given $y$; averaging over the common age must be done after forming the conditional pair <probabilities>. The <continuous-birth binary state fractions> below are consequently volumes in this cube, not products of the marginal single-star fractions.
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