Question

If two events A and B are such that $P ( A + B ) = \frac { 5 } { 6 }$ $P ( A B ) = \frac { 1 } { 3 }$ and $P ( \bar { A } ) = \frac { 1 } { 2 }$ then the events A and B are

• A. Independent
• B. Mutually exclusive
• C. Mutually exclusive and independent
• D. None of these

Answer 1

Answer: (A)

Independent

Answer 2

We have $P ( A + B ) = P ( A ) + P ( B ) - P ( A B )$

$\Rightarrow \frac { 5 } { 6 } = \frac { 1 } { 2 } + P ( B ) - \frac { 1 } { 3 } \Rightarrow P ( B ) = \frac { 4 } { 6 } = \frac { 2 } { 3 }$

Thus, $P ( A ) \cdot P ( B ) = \frac { 1 } { 2 } \times \frac { 2 } { 3 } = \frac { 1 } { 3 } = P ( A B )$

Hence events $A$ and are independent.