Complement and union/intersection laws

5 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = 0.75\).

1. Find \(\mathrm { P } ( A \cap B )\) and \(\mathrm { P } ( A \cup B )\).
2. Determine, giving a reason in each case,
   (a) whether \(A\) and \(B\) are mutually exclusive,
   (b) whether \(A\) and \(B\) are independent.
3. 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 \(\mathrm { P } ( C )\).

3. The probability that Ajita gets up before 6.30 am in the morning is 0.7 The probability that she goes for a run in the morning is 0.35
The probability that Ajita gets up after 6.30 am and does not go for a run is 0.22
Let \(A\) represent the event that Ajita gets up before 6.30 am and \(B\) represent the event that she goes for a run in the morning. Find

1. \(\mathrm { P } ( A \cup B )\),
2. \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\),
3. \(\mathrm { P } ( B \mid A )\).
4. State, with a reason, whether or not events \(A\) and \(B\) are independent.