1. Use the formula for union:

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

Given:

$P(A)=\frac{1}{3},\quad P(B)=\frac{1}{5},\quad P(A\cup B)=\frac{1}{2}$

So,

$\frac{1}{2}=\frac{1}{3}+\frac{1}{5}-P(A\cap B)$

$P(A\cap B)=\frac{1}{3}+\frac{1}{5}-\frac{1}{2}$

Taking LCM $30$,

$P(A\cap B)=\frac{10+6-15}{30}=\frac{1}{30}$

2. Find $P(A\mid B')$:

$P(A\mid B')=\frac{P(A\cap B')}{P(B')}$

Now,

$P(A\cap B')=P(A)-P(A\cap B)=\frac{1}{3}-\frac{1}{30}=\frac{10-1}{30}=\frac{9}{30}=\frac{3}{10}$

and

$P(B')=1-P(B)=1-\frac{1}{5}=\frac{4}{5}$

Hence,

$P(A\mid B')=\frac{\frac{3}{10}}{\frac{4}{5}}=\frac{3}{10}\cdot\frac{5}{4}=\frac{3}{8}$

3. Find $P(B\mid A')$:

$P(B\mid A')=\frac{P(B\cap A')}{P(A')}$

Now,

$P(B\cap A')=P(B)-P(A\cap B)=\frac{1}{5}-\frac{1}{30}=\frac{6-1}{30}=\frac{5}{30}=\frac{1}{6}$

and

$P(A')=1-P(A)=1-\frac{1}{3}=\frac{2}{3}$

Hence,

$P(B\mid A')=\frac{\frac{1}{6}}{\frac{2}{3}}=\frac{1}{6}\cdot\frac{3}{2}=\frac{1}{4}$

4. Add the two values:

$P(A\mid B')+P(B\mid A')=\frac{3}{8}+\frac{1}{4}$

$=\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$

5. Compare with options:

$\frac{5}{8}$

So the correct option is B.