A problem in Mathematics is given to two students X and Y whose chances of solving it are \(\displaystyle\frac{1}{3}\) and \(\displaystyle\frac{1}{4}\) respectively. The probability that only X solves the problem, is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{4}\)
“Only X solves” means X solves and Y does not. The two students work independently, so the probabilities multiply:
\( \displaystyle P(\text{only X}) = P(X)\,P(\overline{Y}) = \frac{1}{3}\left(1 - \frac{1}{4}\right) \)
\( \displaystyle = \frac{1}{3} \times \frac{3}{4} = \frac{1}{4} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{3}\) — Incorrect. This is the probability that X solves it, which also includes the cases where Y solves it too.
Option (2) \(\displaystyle\frac{1}{12}\) — Incorrect. This is \(\frac{1}{3} \times \frac{1}{4}\), the probability that both solve it.
Option (4) \(\displaystyle\frac{7}{12}\) — Incorrect. This is the sum \(\frac{1}{3} + \frac{1}{4}\), which is not the probability of any event here; even “at least one solves” is \(1 - \frac{2}{3} \times \frac{3}{4} = \frac{1}{2}\).