2025 VJC P2 Q6

Two events \(A\) and \(B\) are such that \(\mathrm{P}(A) = p\) and \(\mathrm{P}(B) = \frac{5}{4}p\). It is given that \(\mathrm{P}(A \cup B) = \mathrm{P}(A|B) = 0.6\).

1. Find the value of \(p\).[3]
2. Explain whether the events \(A'\) and \(B\) are independent.[1]

The events \(C\) and \(D\) are such that \(\mathrm{P}(C) = 0.55\), \(\mathrm{P}(D) = 0.6\) and \(\mathrm{P}(A \cap C) = \mathrm{P}(A \cap D) = \mathrm{P}(C \cap D) = q\).

3. Find the value of \(q\) that gives the minimum value of \(\mathrm{P}(A \cup C \cup D)\) and state the minimum value of \(\mathrm{P}(A \cup C \cup D)\).[2]