Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 12 / d / Solution

Let $X_1$ and $X_2$ denote the first and second labels. Sampling with replacement restores the bag before the second draw, so conditioning on $X_1=x$ only specifies the threshold. For $1\le x\le g$, there are $x$ qualifying green balls and $x$ qualifying red balls. For $g<x\le r$, every green ball and the first $x$ red balls qualify. Hence the requested conditional probability isThe formulas agree at $x=g$ and give one at $x=r$. Labels outside the possible first-draw range do not define a conditioning event of positive probability.