$\begin {array} {1 1} (A)\;\large\frac{1}{2} & \quad (B)\;\large\frac{3}{4} \\ (C)\;\large\frac{2}{3} & \quad (D)\;None\:of \: these \end {array}$

Let $ P(A) = $ Probability that A will hit B

Let $ P(B) = $ Probability that B will hit A

Let $ P(C) = $ Probability that C will hit A

Let $ P(E) = $ Probability that A will be hit.

Then

$P(E) = 1-P ( \overline B \cap \overline C )$

$ = 1- P( \overline B ). P ( \overline C )$

$ 1- \large\frac{1}{2}. \large\frac{2}{3}$

$ = \large\frac{2}{3}$

$ \Rightarrow P \bigg( \large\frac{ B \cap \overline C }{E} \bigg) = \large\frac{P(B).P( \overline C )}{ P(E)}$

$ \Large\frac{ \Large\frac{1}{2}.\Large\frac{1}{2}}{ \Large\frac{2}{3}} = \large\frac{1}{2}$

Hence Ans (A)

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