2022 VJC P2 Q5

For events \(A\) and \(B\), it is given that \(\mathrm{P}\left( A \right)=0.6\), \(\mathrm{P}\left( B \right)=0.2\) and \(\mathrm{P}\left( \left. A \right|B' \right)=0.55\). Find

1. \(\mathrm{P}\left( A\cap B' \right)\).[1]
2. \(\mathrm{P}\left( A'\cap B' \right)\).[2]

For a third event \(C\), it is given that \(\mathrm{P}\left( C \right)=0.4\), \(\mathrm{P}\left( A\cap C \right)=\mathrm{P}\left( B\cap C \right)\), \(\mathrm{P}\left( A'\cap B'\cap C \right)=0.24\) and \(\mathrm{P}\left( A\cap B\cap C \right)=0.1\).

Determine whether \(A\) and \(C\) are independent.[3]