Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 16 / c / ii / Solution

There are ten balls of each colour. Under uniform sampling without replacement, all $\left(\genfrac{}{}{0ex}{}{20}{12}\right)$ subsets of twelve labelled balls are equally likely. A favourable subset chooses four of the ten white balls and eight of the ten black balls, giving $\left(\genfrac{}{}{0ex}{}{10}{4}\right)\left(\genfrac{}{}{0ex}{}{10}{8}\right)$ choices. The hypergeometric distribution therefore givesThus the requested integers are $\boxed{N_1=N_2=10,m_1=4,m_2=8}$, with $N=20$ and $k=12$.