$A$ is targetting to $B, B$ and $C$are targetting to $A$.The probability of hitting the target by $A, B$and $C$ are $\frac{2}{3},\frac{1}{2},\frac{1}{3}$ respectively. If $A$is hit, the probability, that $B$ hits the target and $C$ does not, is

$P\left(A\right)=$probability that $A$ will hit $B=\frac{2}{3}$

$P\left(B\right)=$probability that B will hit $A=\frac{1}{2}$

$P\left(C\right)=$probability that C will hit $A=\frac{1}{3}$

$P\left(E\right)=$probability that A will be hit $=1-P\left(\overline{B}\right)\cdot P\left(\overline{C}\right)$

$=1-\frac{{\displaystyle 1}}{{\displaystyle 2}}\frac{{\displaystyle 2}}{{\displaystyle 3}}=\frac{{\displaystyle 2}}{{\displaystyle 3}}$ [since $E=\mid (B\cup C)$]

Now $P\left(\frac{B\cap \overline{C}}{E}\right)=\frac{P(B\cap \overline{C}\cap (B\cup C\left)\right)}{P\left(E\right)}$

$=\frac{P(B\cap \overline{C})}{P\left(E\right)}=\frac{P\left(B\right)\cdot P\left(\overline{C}\right)}{P\left(E\right)}=\raisebox{1ex}{$\frac{1}{2}\cdot \frac{2}{3}$}\!\left/ \!\raisebox{-1ex}{$\frac{2}{3}$}\right.=\frac{1}{2}$