5 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = 0.75\).
- Find \(\mathrm { P } ( A \cap B )\) and \(\mathrm { P } ( A \cup B )\).
- Determine, giving a reason in each case,
(a) whether \(A\) and \(B\) are mutually exclusive,
(b) whether \(A\) and \(B\) are independent. - A further event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.05\). It is also given that \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C \right) = \mathrm { P } \left( A ^ { \prime } \cap B \cap C \right) = x\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 2 x\).
Find \(\mathrm { P } ( C )\).