2025 DHS P2 Q6

The events \(A\), \(B\) and \(C\) are such that \(\mathrm{P}(A)=0.8\), \(\mathrm{P}(B)=0.2\) and \(\mathrm{P}(C)=0.6\). It is also known that \(\mathrm{P}(A \cap B)=\mathrm{P}(B)\) and \(\mathrm{P}(B \cap C)=0.1\).

1. Find exactly the maximum and minimum possible values of
   1. \(\mathrm{P}(A \cap B' \cap C)\),[3]
   2. \(\mathrm{P}(A \cap B' \cap C | A \cup C)\).[2]
2. It is given further that \(A\) and \(C\) are independent, find the value of \(\mathrm{P}(A \cap B' \cap C)\).[2]