Probabilities to solve a specific problem by A, B and C are \(\displaystyle\frac{1}{2}, \frac{1}{3}\) and \(\displaystyle\frac{1}{4}\) respectively. Probability that at least one will solve the problem is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{3}{4}\)
A, B and C try the problem independently. “At least one solves it” is the complement of “none of them solves it”.
Probabilities of not solving:
\( \displaystyle 1 - \frac{1}{2} = \frac{1}{2},\quad 1 - \frac{1}{3} = \frac{2}{3},\quad 1 - \frac{1}{4} = \frac{3}{4} \)
Probability that none solves it:
\( \displaystyle \frac{1}{2}\times\frac{2}{3}\times\frac{3}{4} = \frac{1}{4} \)
Hence
\( \displaystyle P(\text{at least one solves}) = 1 - \frac{1}{4} = \frac{3}{4} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{24}\) — Incorrect. This is the probability that all three solve it: \(\frac{1}{2}\times\frac{1}{3}\times\frac{1}{4} = \frac{1}{24}\).
Option (2) \(\displaystyle\frac{1}{4}\) — Incorrect. This is the probability that none of them solves it.
Option (3) \(\displaystyle\frac{23}{24}\) — Incorrect. This is \(1 - \frac{1}{24}\), the probability that not all three solve it.