Let \(A\) and \(B\) be any two events. Define the new events \(C\), \(\hat A\), and \(\hat B\) by \(C=A\cap B\), \(\hat A=A \cap B^c\), and \(\hat B = B \cap A^c\) where \(A^c\) is the compliment of \(A\) and \(B^c\) is the compliment of \(B\).

- Argue that \(A \cup B = \hat A \cup \hat B \cup C\) and that all three sets are mutually disjoint. i.e. \(\hat A\cap C = \emptyset\), \(\hat B\cap C = \emptyset\), and \(\hat A\cap \hat B = \emptyset\).
- Show that \(\mathbf{P}(A)= \mathbf{P}(\hat A) + \mathbf{P}(C)\) and \(\mathbf{P}(B)= \mathbf{P}(\hat B) + \mathbf{P}(C)\) .
- Show that \(\mathbf{P}(A \cup B) = \mathbf{P}(A) + \mathbf{P}(B) – \mathbf{P}(A \cap B)\).