Evaluate

It is given that, $2 P(A)=P(B)=\frac{5}{13}$

$\Rightarrow P(A)=\frac{5}{26}$ and $P(B)=\frac{5}{13}$

$P(A \mid B)=\frac{2}{5}$

$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}=\frac{2}{5}$

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

It is known that, $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$

$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{26}+\frac{5}{13}-\frac{2}{13}$

$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5+10-4}{26}$

$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{11}{26}$