If A and B are events such that $P(A' ∪B')=\frac{1}{3} $ and $P(A ∪ B)=\frac{4}{9}$ then the value of $P(A')+P(B')$ is :

$\frac{8}{9}$

The correct answer is Option (3) → $\frac{8}{9}$

$P(\overline{A}∪\overline{B})=P(\overline{A∩B})=1-P(A∩B)$

so $P(A∩B)=\frac{2}{3}$

so $P(A∪B)=\frac{4}{9}⇒P(\overline{A∪B})=\frac{5}{9}$

$⇒P(\overline{A}∩\overline{B})=\frac{5}{9}$

so $P(\overline{A})+P(\overline{B})=P(\overline{A}∪\overline{B})+P(\overline{A}∩\overline{B})=\frac{8}{9}$