2020 EJC P2 Q7

For events \(A\) and \(B\), it is given that \(\mathrm{P}\left( A\cap {B}' \right)=0.32\) and \(\mathrm{P}\left( B|A \right)=\frac{7}{23}\).

1. Show that \(\mathrm{P}\left( A\cap B \right)=0.14\).[2]
2. Given further that \(\mathrm{P}\left( A|B \right)=\frac{7}{19}\), find \(\mathrm{P}\left( A\cup B \right)\).[2]

It is given that \(\mathrm{P}\left( C \right)=0.5\), where \(C\) is an event such that events \(A\) and \(C\) are independent and \(B\) and \(C\) are independent.

3. Find the greatest and least possible values of \(P\left( {A}'\cap {B}'\cap {C}' \right)\).[4]