Question

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

Answer 1

It is given that $P(A)=\frac{1}{2},P(B)=\frac{7}{12}$ and $P$(not $A$ or not $B$)$=\frac{1}{4}$
$\implies P(A'\cup B')=\frac{1}{4}$
$\implies P\bigg((A\cap B)'\bigg)=\frac{1}{4}$
$⇒\frac{1}{4}=1-P(A∩B)$
$⇒P(A∩B)=\frac{3}{4}$
However, P(A).P(B)$$=\frac{1}{2}×\frac{7}{12}=\frac{7}{24}
$\therefore$ $P(A∩B)≠P(A).P(B)$ ,i.e., $A$ and $B$ are not independent.