2022 DHS J2 MYE P2 Q2

For independent events \(A\) and \(B\), it is given that \(\mathrm{P}\left( A \right)=0.5\) and \(\mathrm{P}\left( B \right)=0.45\).

1. Prove that \(A'\) and \(B\) are independent.[2]

For a third event \(C\), it is given that \(\mathrm{P}\left( C \right)=0.4\).

2. If \(A\) and \(C\) are not independent and \(\mathrm{P}\left( A|C \right)=0.3\), find \(\mathrm{P}\left( A\cup C \right)\).[2]
3. If \(A\) and \(C\) are independent, find the maximum and minimum possible values of \(\mathrm{P}\left( A'\cap B\cap C' \right)\).[4]