2023 EJC P2 Q6

For events \(A\) , \(B\) and \(C\) it is given that \(\mathrm{P}\left( A \right)\mathrm{=P}\left( A\cap B \right)=\frac{1}{5}\) and \(\mathrm{P}\left( B \right)\mathrm{=2P}\left( A \right)\). It is also given that \(\mathrm{P}\left( B\cap C \right)=\frac{3}{10}\).

1. Find the greatest and least possible values of \(\mathrm{P}\left( A\cap C \right)\).[4]
2. Draw a Venn diagram showing all \(3\) events representing the case where the value of \(\mathrm{P}\left( A\cap C \right)\) is the greatest.[1]

It is now known that \(\mathrm{P}\left( A'\cap B'\cap C' \right)=\frac{1}{12}\).

3. If \(A\) and \(C\) are independent events, calculate the value of \(\mathrm{P}\left( A\cap C \right)\).[2]