Question

A and B are two independent events. The probability that both A and B occur is $\frac { 1 } { 6 }$ and the probability that neither of them occurs is $\frac { 1 } { 3 }$ . Then the probability of the two events are respectively

• A. $\frac { 1 } { 2 }$and $\frac { 1 } { 3 }$
• B. $\frac { 1 } { 5 }$and $\frac { 1 } { 6 }$
• C. $\frac { 1 } { 2 }$and $\frac { 1 } { 6 }$
• D. $\frac { 2 } { 3 }$ and $\frac { 1 } { 4 }$

Answer 1

Answer: (A)

$\frac { 1 } { 2 }$and $\frac { 1 } { 3 }$

Answer 2

$P ( A \cap B ) = P ( A ) \cdot P ( B ) = \frac { 1 } { 6 }$

$P ( \bar { A } \cap \bar { B } ) = \frac { 1 } { 3 } = 1 - P ( A \cup B )$

$\Rightarrow \frac { 1 } { 3 } = 1 - [ P ( A ) + P ( B ) ] + \frac { 1 } { 6 } \Rightarrow P ( A ) + P ( B ) = \frac { 5 } { 6 }$

Hence $P ( A )$ and $P ( B )$ are $\frac { 1 } { 2 }$ and $\frac { 1 } { 3 }$