Evaluate $(A \cup B)$ , if $2P(A) = P(B) = \cfrac{5}{{13}}$ and $P(A|B) = \cfrac{2}{5}$

.: Given, $P(A|B) = \cfrac{2}{5} \Rightarrow \cfrac{{P(A \cap B)}}{{P(B)}} = \cfrac{2}{5}$

$\Rightarrow P(A \cap B) = \cfrac{2}{5}P(B) = \cfrac{2}{5} \times \cfrac{5}{{13}} = \cfrac{2}{{13}}$

Hence, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$= \cfrac{1}{2} \times \cfrac{5}{{13}} + \cfrac{5}{{13}} - \cfrac{2}{{13}}$

$= \cfrac{{5 + 10 - 4}}{{26}} = \cfrac{{11}}{{26}}$