2019 JPJC P2 Q11

[Leave your answers in fraction]

The events \(A\) and \(B\) are such that \(\mathrm{P}\left( A|B \right)=\frac{7}{10}\), \(\mathrm{P}\left( B|A \right)=\frac{4}{15}\) and \(\mathrm{P}\left( A\cup B \right)=\frac{3}{5}\).

Find the exact values of

1. \(\mathrm{P}\left( A\cap B \right)\),[3]
2. \(\mathrm{P}\left( A'\cap B \right)\).[2]

Find a third event \(C\), it is given that \(\mathrm{P}\left( C \right)=\frac{3}{10}\) and that \(A\) and \(C\) are independent.

3. Find \(\mathrm{P}\left( A'\cap C \right)\).[2]
4. Hence state an inequality satisfied by \(\mathrm{P}\left( A'\cap B\cap C \right)\).[1]