The probabilities that A and B can solve a problem independently are \(\displaystyle\frac{1}{4}\) and \(\displaystyle\frac{2}{5}\) respectively. If both try the problem independently, the probability that the problem is solved is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{11}{20}\)
The problem is solved if at least one of them solves it. It is easier to use the complement: the problem stays unsolved only when both fail.
\( \displaystyle P(A \text{ fails}) = 1-\frac{1}{4} = \frac{3}{4} \)
\( \displaystyle P(B \text{ fails}) = 1-\frac{2}{5} = \frac{3}{5} \)
Since they work independently, the failures multiply:
\( \displaystyle P(\text{both fail}) = \frac{3}{4}\times\frac{3}{5} = \frac{9}{20} \)
\( \displaystyle P(\text{solved}) = 1-\frac{9}{20} = \frac{11}{20} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{13}{20}\) — Incorrect. This is \(\frac{1}{4}+\frac{2}{5}\), which counts the case when both solve it twice.
Option (3) \(\displaystyle\frac{9}{20}\) — Incorrect. This is the probability that both fail, so the problem is not solved.
Option (4) \(\displaystyle\frac{1}{10}\) — Incorrect. This is \(\frac{1}{4}\times\frac{2}{5}\), the probability that both solve it.