Read the text carefully and answer the questions:
Three persons A, B and C were given a task, whose probabilities of completing the task on time are \(\displaystyle\frac{1}{3}, \frac{1}{4}\) and \(\displaystyle\frac{1}{5}\) respectively. They were asked to complete the task on time independently.
The probability that exactly one of them completes the task on time is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{13}{30}\)
Exactly one completes on time in three mutually exclusive ways; in each, the other two do not. With \(P(\bar{A}) = \frac{2}{3}\), \(P(\bar{B}) = \frac{3}{4}\), \(P(\bar{C}) = \frac{4}{5}\):
\( \displaystyle P(A\cap\bar{B}\cap\bar{C}) = \frac{1}{3}\times\frac{3}{4}\times\frac{4}{5} = \frac{12}{60} \)
\( \displaystyle P(\bar{A}\cap B\cap\bar{C}) = \frac{2}{3}\times\frac{1}{4}\times\frac{4}{5} = \frac{8}{60} \)
\( \displaystyle P(\bar{A}\cap\bar{B}\cap C) = \frac{2}{3}\times\frac{3}{4}\times\frac{1}{5} = \frac{6}{60} \)
Adding:
\( \displaystyle P(\text{exactly one}) = \frac{12 + 8 + 6}{60} = \frac{26}{60} = \frac{13}{30} \)