1 For the mutually exclusive events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x\), where \(x \neq 0\).
- Show that \(x \leqslant \frac { 1 } { 2 }\).
- Show that \(A\) and \(B\) are not independent.
The event \(C\) is independent of \(A\) and also independent of \(B\), and \(\mathrm { P } ( C ) = 2 x\).
- Show that \(\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )\).