QJEEM17Maf8d9: JEE Main 2017 Mathematics - Class 12 JEEM17Maf8d9 Solution: Three persons P, Q and R independently try to hit a target. I…

We have the following probabilities:

$\bullet$ The probability that the target is hit by the person P is ${3 \over 4}$ .

$\bullet$ The probability that the target is not hit by the person P is $1 - {3 \over 4} = {1 \over 4}$ .

$\bullet$ The probability that the target is hit by the person Q is ${1 \over 2}$ .

$\bullet$ The probability that the target is not hit by the person Q is $1 - {1 \over 2} = {1 \over 2}$ .

$\bullet$ The probability that the target is hit by the person R is ${5 \over 8}$ .

$\bullet$ The probability that the target is not hit by the person R is $1 - {5 \over 8} = {3 \over 8}$ .

Here, we have used the fact that if the probability of occurrence of an event is p, then the probability of non-occurrence of an event is $q = 1 - p$ .

Therefore, the probability that the target is hit by P or Q and not by R is

(Probability that the target is hit by P and not by Q and R) + (Probability that the target is hit by Q and not by P and R) + (Probability that the target is hit by both P and Q and not by R)

$= \left( {{3 \over 4}} \right)\left( {{1 \over 2}} \right)\left( {{3 \over 8}} \right) + \left( {{1 \over 4}} \right)\left( {{1 \over 2}} \right)\left( {{3 \over 8}} \right) + \left( {{3 \over 4}} \right)\left( {{1 \over 2}} \right)\left( {{3 \over 8}} \right)$

$= {9 \over {64}} + {3 \over {64}} + {9 \over {64}} = {{9 + 3 + 9} \over {64}} = {{21} \over {64}}$