2023 RI P2 Q7

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

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

It is given that \(\mathrm{P}\left( A\cup B \right)=0.7\) and \(b=0.5\).

2. Show that \(a=0.4\).[2]

For a third event \(C\), it is given that \(\mathrm{P}\left( C \right)=0.3\) and events \(A\) and \(C\) are independent.

3. If events \(B\) and \(C\) are mutually exclusive, find \(\mathrm{P}\left( A'\cap B'\cap C' \right)\).[2]
4. If events \(B\) and \(C\) are not mutually exclusive, find the greatest possible value of \(\mathrm{P}\left( A'\cap B'\cap C' \right)\), showing your working clearly.[3]