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

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

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)