2018 IJC P2 Q8

For events \(A\), \(B\), and \(C\), it is given that \(\mathrm{P}\left( A \right)=0.5\), \(\mathrm{P}\left( B \right)=0.45\), and \(\mathrm{P}\left( C \right)=0.35\).

It is further given that \(\mathrm{P}\left( B\left| C \right. \right)=0.5\), \(\mathrm{P}\left( A\cap C \right)=0.15\), and \(\mathrm{P}\left( A\cap B\cap C \right)=0.1\).

1. Find \(\mathrm{P}\left( {A}'\cap B\cap C \right)\).[2]
2. Given also that events \(A\) and \(B\) are independent, find \(\mathrm{P}\left( A\cup B \right)\)[2]
3. Given instead that events \(A\) and \(B\) are not independent, find the greatest and least possible values of \(\mathrm{P}\left( {A}'\cap {B}'\cap {C}' \right)\).[4]