3 For the events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = \frac { 3 } { 4 }\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = \frac { 1 } { 2 }\).
- Find \(\mathrm { P } ( A \cap B )\).
For a third event \(C , \mathrm { P } ( C ) = \frac { 1 } { 4 }\) and \(C\) is independent of the event \(A \cap B\).
- Find \(\mathrm { P } ( A \cap B \cap C )\).
- Given that \(\mathrm { P } ( C \mid A ) = \lambda\) and \(\mathrm { P } ( B \mid C ) = 3 \lambda\), and that no event occurs outside \(A \cup B \cup C\), find the value of \(\lambda\).