Question

If A and B are two independent events, then A and $\bar { B }$ are

• A. Not independent
• B. Also independent
• C. Mutually exclusive
• D. None of these

Answer 1

Answer: (B)

Also independent

Answer 2

Since $A \cap \bar { B }$ and $A \cap B$ are mutually exclusive events such that $A = ( A \cap \bar { B } ) \cup ( A \cap B )$

∴ $P ( A ) = P ( A \cap \bar { B } ) + P ( A \cap B )$

$\Rightarrow P ( A \cap \bar { B } ) = P ( A ) - P ( A \cap B )$ $= P ( A ) - P ( A ) P ( B )$

$( \because A , B$ are independent)

$\Rightarrow P ( A \cap \bar { B } )$ $= P ( A ) ( 1 - P ( B ) ) = P ( A ) P ( \bar { B } )$

$\therefore A$ and $\bar { B }$ are also independent.