2021 EJC P2 Q6

For events \(A\) and \(B\) it is given that \(\mathrm{P}\left( A \right)=\frac{3}{5}\), \(\mathrm{P}\left( A\cap B \right)=\frac{7}{60}\) and \(\mathrm{P}\left( \left. A' \right|B \right)=\frac{13}{20}\).

1. Show that \(\mathrm{P}\left( B \right)=\frac{1}{3}\).[3]

For a third event \(C\), it is given that \(\mathrm{P}\left( C \right)=\frac{3}{10}\) and that \(\mathrm{P}\left( B\cap C \right)=\frac{1}{10}\).

1. Determine if events \(B\) and \(C\) are independent.[1]
2. Given also that \(\mathrm{P}\left( A\cap B\cap C \right)=\frac{1}{15}\), find the greatest and least possible values of \(\mathrm{P}\left( A\cap B'\cap C \right)\).[3]