A problem in mathematics is given to three students whose chances of solving it independently are 1/2, 1/3 and 1/4 respectively. The probability that the problem is solved is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3/4
The problem is solved if at least one student solves it, so the complement is that nobody solves it. The students work independently, so the probabilities of failure multiply.
\( \displaystyle P(\text{no one solves})=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right) \)
\( \displaystyle =\frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}=\frac{1}{4} \)
\( \displaystyle P(\text{problem is solved})=1-\frac{1}{4}=\frac{3}{4} \)
Why the other options are incorrect
Option (1) 1/24 — Incorrect. \( \frac{1}{24}=\frac{1}{2}\times\frac{1}{3}\times\frac{1}{4} \) is the probability that all three solve it, not that at least one does.
Option (2) 1/6 — Incorrect. The probability is 3/4, not 1/6.
Option (4) 1/12 — Incorrect. The probability is 3/4, not 1/12.