Events $A$ and $B$ are such that $P(A) = \cfrac{1}{2},P(B) = \cfrac{7}{{12}}$ and $P({\rm{ not }}A{\rm{ or not }}B) = \cfrac{1}{4}$ State whether $A$ and $B$ are independent?

.: Given that $P$ (not $A$ or not $B$ ) $= \cfrac{1}{4} \Rightarrow P\left( {{A^\circ } \cup {B^c}} \right) = \cfrac{1}{4}$

$\Rightarrow P\left( {{{(A \cap B)}^c}} \right) = \cfrac{1}{4}$

$\Rightarrow 1 - P(A \cap B) = \cfrac{1}{4}$

$\Rightarrow P({\rm{A}} \cap B) = 1 - \cfrac{1}{4} = \cfrac{3}{4}$

Also, $P(A) \times P(B) = \cfrac{1}{2} \times \cfrac{7}{{12}} = \cfrac{7}{{24}} \ne \cfrac{3}{4}$ $\Rightarrow$ $P(A) \times P(B) \ne P(A \cap B)A$

$\therefore$ $A$ and $B$ are not independent.