NIMCET 2015 — Mathematics PYQ

Let $P(A),P(B),$ and $P(C)$ be the probabilities of A, B, and C hitting their respective targets.

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

$P(B)=\frac{1}{2}⟹P(B^{\prime })=1-\frac{1}{2}=\frac{1}{2}$

$P(C)=\frac{1}{3}⟹P(C^{\prime })=1-\frac{1}{3}=\frac{2}{3}$

A is hit if either B hits, C hits, or both hit.

$P(\text{A is hit})=1-P(\text{B misses and C misses})$

$P(\text{A is hit})=1-(P(B^{\prime })\times P(C^{\prime }))$

$P(\text{A is hit})=1-(\frac{1}{2}\times \frac{2}{3})$

$P(\text{A is hit})=1-\frac{1}{3}=\frac{2}{3}$

We need to find the conditional probability that B hit and C did not, given that A was hit:

$P(B\cap C^{\prime }|\text{A is hit})=\frac{P(B\cap C^{\prime })}{P(\text{A is hit})}$

Since B and C are independent:

$P(B\cap C^{\prime })=P(B)\times P(C^{\prime })=\frac{1}{2}\times \frac{2}{3}=\frac{1}{3}$

Required Probability:

$P=\frac{1/3}{2/3}$

$P=\frac{1}{2}$

Correct Option: (a)