If $P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{5};$ A and B are independent events then $P\left(\frac{A}{A\cup B}\right)$ =

$P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{5};$A,B are independent events

$P\left(A\cap B\right)=P\left(A\right)P\left(B\right)=\frac{1}{2}\times \frac{1}{5}=\frac{1}{10}$

$P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)=\frac{1}{2}+\frac{1}{5}-\frac{1}{10}=\frac{6}{10}=\frac{3}{5}$

$P\left(\frac{A}{A\cup B}\right)=\frac{P\left(A\cap \left(A\cup B\right)\right)}{P\left(A\cup B\right)}=\frac{P\left(A\right)}{P\left(A\cup B\right)}=\frac{1/2}{3/5}=\frac{5}{6}$