2015 DHS P2 Q9

For independent events \(A\) and \(B\), it is given that \(\mathrm{P}(A\cap B)=0.05\) and \(\mathrm{P}\left( A\cup B \right)=0.55\).

1. Find \(\mathrm{P}\left( A \right)\) and \(\mathrm{P}\left( B \right)\) if \(\mathrm{P}\left( A \right)<\mathrm{P}\left( B \right)\).[4]

For a third event \(C\), \(\mathrm{P}\left( C \right)=0.5\) and \(\mathrm{P}\left( A'\cup B'|C \right)=0.95\).

2. Find \(\mathrm{P}\left( A\cap B\cap C \right)\).[3]
3. State, with reason, whether the events \(A\)  and \(C\) are mutually exclusive.[1]