2021 VJC P2 Q7

For events \(A\),\(B\) and \(C\) it is given that \(\mathrm{P}\left( A \right)=0.8\), \(\mathrm{P}\left( B \right)=0.4\), \(\mathrm{P}\left( C \right)=c\) and \(\mathrm{P}\left( A\cup B \right)=0.95\).

1. Find \(\mathrm{P}\left( A\cap B \right)\) and determine with reason, if events \(A\) and \(B\) are independent.[2]
2. State the range of values of \(c\) which necessarily implies that events \(A\) and \(C\) are not mutually exclusive.[1]

It is now given that \(\mathrm{P}\left( C \right)=\mathrm{P}\left( \left. C \right|A \right)=0.45\).

3. Find \(\mathrm{P}\left( A\cap C \right)\) and state the greatest and least possible values of \(\mathrm{P}\left( A'\cap B\cap C \right)\).[3]