JEE Main 2017 — Probability Question with Solution

We have the following probabilities:

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

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

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

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

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

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

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)

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

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