Browse Questions

# If A and B are two events such that $P ( A \cup B) = \large\frac{5}{6}, \: P ( A \cap B ) = \large\frac{1}{3} \: and \: P(B') = \large\frac{1}{2}$ then events A and B are

$\begin {array} {1 1} (1)\;dependent & \quad (2)\;independent \\ (3)\;mutually\: exclusive & \quad (4)\;none\: of \: these \end {array}$

$P (B') = \large\frac{1}{2}$
$\Rightarrow P (B) = \large\frac{1}{2}$
$P ( A \cup B ) = \large\frac{5}{6}$
$\Rightarrow P ( A) + P (B) - P ( A \cap B ) = \large\frac{5}{6}$
$\Rightarrow P (A) + \large\frac{1}{2}-\large\frac{1}{3}=\large\frac{5}{6}$
$P(A) = \large\frac{2}{3}$
Now $P(A)\: P(B)= \bigg( \large\frac{ \not{2} }{3} \bigg) \bigg( \large\frac{1}{ \not{2} } \bigg)$
$= \large\frac{1}{3}$
$P ( A \cap B ) = \large\frac{1}{3}$
$\therefore$ A and B are independent