Prove independence relationship algebraically

A question is this type if and only if it asks to show or prove that events cannot be independent (or must be dependent) using algebraic manipulation of given probability constraints.

1 questions

OCR S4 2008 June Q1
1 For the mutually exclusive events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x\), where \(x \neq 0\).
  1. Show that \(x \leqslant \frac { 1 } { 2 }\).
  2. Show that \(A\) and \(B\) are not independent. The event \(C\) is independent of \(A\) and also independent of \(B\), and \(\mathrm { P } ( C ) = 2 x\).
  3. Show that \(\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )\).