$P(A'\cup B')=\large\frac{2}{3}$
$\Rightarrow P(\overline{ A\cap B})=\large\frac{2}{3}$
$1-P(A\cap B)=\large\frac{2}{3}$
$P(A\cap B)=1-\large\frac{2}{3}=\frac{1}{3}$
Also $P(A\cup B)=\large\frac{5}{9}$
$P(A)+P(B)-P(A\cap B)+P(A\cup B)$
$1-P(\bar{A})+1-P(A\cap B)=P(A\cup B)$
$P(\bar{A})+P(\bar{B})=2-P(A\cap B)=P(A\cup B)$
=$\Large 2-\frac{5}{9}-\frac{1}{3}=\frac{10}{9}$
$P(\bar{A})+P(\bar{B})=\large\frac{10}{9}$