Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 42 / 3 / iii / Solution

For continuous-birth binary state fractions, conditional mass independence and the common age giveThe factor two in the middle expression allows either component to be the red giant. ThusGeometrically, the corresponding sections of the unit cube have areas proportional to $y^2$, so their volumes equal one third of their $y=1$ base areas. This is exactly the pyramid-volume calculation: the equal height and common factor $1/3$ leave the base-area ratio $1:8:16$ unchanged.