Probabilities of solving a specific mathematical problem independently by three students are \( \displaystyle\frac{1}{2},\displaystyle\frac{1}{3} \) and \( \displaystyle\frac{1}{4} \) . If all the three students try to solve the problem independently, the probability that the problem is solved, is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{3}{4} \)
The problem is solved if at least one student solves it. It stays unsolved only if all three fail, and the students work independently, so the complement is the quickest route.
\( \displaystyle P(\text{all fail})=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right) \)
\( \displaystyle =\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4} \)
\( \displaystyle =\frac{1}{4} \)
\( \displaystyle P(\text{solved})=1-\frac{1}{4}=\frac{3}{4} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{1}{24} \) — Incorrect. \( \frac{1}{24}=\frac{1}{2}\cdot\frac{1}{3}\cdot\frac{1}{4} \) is the probability that all three students solve it, not that at least one does.
Option (2) \( \displaystyle\frac{1}{12} \) — Incorrect. It is less than \( \frac{1}{2} \), the chance that the first student alone solves it, so it is too small.
Option (4) 1 — Incorrect. 1 would mean the problem is certainly solved, but all three fail with probability \( \frac{1}{4} \).