UPSC NDA Math Model Paper 2

It is given that P(A) = $\frac{1}{2}$ and P(not B) = $\frac{1}{4}$ . Let's say that the probability of C solving the problem is P(C) = x. $\therefore$ Probability of A not solving the problem $=P(\text{not } A)=1-P(A)=1-\frac{1}{2}=\frac{1}{2}$. And, probability of C not solving the problem = P(not C) = 1 - P(C) = 1 - x. Now, the probability that the problem is not solved at all: = P(not A) AND P(not B) AND P(not C) = P(not A) × P(not B) × P(not C) $=\frac{1}{2} \times \frac{1}{4} \times (1-x)$ $=\frac{1-x}{8}$ And, the probability that the problem is solved = 1 - Probability that the problem is not solved at all $=\frac{63}{64}.$ $\Rightarrow 1-\frac{1-x}{8}=\frac{63}{64}$ $\Rightarrow \frac{1-x}{8}=1-\frac{63}{64}=\frac{1}{64}$ $\Rightarrow 1-x=\frac{1}{8}$ $\Rightarrow x=1-\frac{1}{8}=\frac{7}{8}$ ∴ The probability of C solving the problem is $P(C) = x =\frac{7}{8}$