2025 Specimen Paper P2 Q8

A bag contains \(3\) blue discs, \(2\) red discs and \(y\) yellow discs. Li chooses \(3\) discs at random from the bag, without replacement.

1. Show that the probability Li chooses \(1\) blue disc, \(1\) red disc and \(1\) yellow disc is \(\frac{36y}{\left( y+5 \right)\left( y+4 \right)\left( y+3 \right)}\).[1]

Li’s discs are replaced in the bag.

Darvina chooses \(3\) discs at random from the bag, without replacement. The random variable \(S\) is the sum of the number of blue discs chosen and twice the number of red discs chosen.

2. Find an expression in terms of \(y\) for \(\mathrm{P}\left( S=3 \right)\).[2]
3. Given that \(\mathrm{P}\left( S=3 \right)=\frac{7}{20}\), calculate the value of \(y\). Hence find the probability distribution of \(S\).[6]