2025 TMJC P2 Q6

For events \(A\), \(B\) and \(C\), it is given that \(\mathrm{P}(A) = 0.3\), \(\mathrm{P}(B) = 0.4\), \(\mathrm{P}(C) = 0.42\), \(\mathrm{P}(A' \cap B \cap C) = x\) and \(\mathrm{P}(A \cap B \cap C) = 0.02\). It is also given that events \(A\) and \(B\) are independent, and that events \(A\) and \(C\) are independent.

1. Find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A \cap C)\).[2]
2. Draw a Venn diagram to represent this situation, showing the probability in each of the eight regions, in terms of \(x\) where necessary.[3]
3. Find the greatest and least possible values of \(\mathrm{P}(A' \cap B' \cap C)\).[2]