Kashvi, Kalyani and Sara of class 12 are given a problem in Accounts whose respective probabilities of solving are \(\displaystyle\frac{2}{5}, \frac{1}{4}\) and \(\displaystyle\frac{1}{6}\). They were asked to solve it independently.
The probability that either only Kalyani or Kashvi or Sara solves it is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{9}{20}\)
“Either only Kalyani or Kashvi or Sara solves it” means that only one of the three solves it, that is, exactly one of them solves the problem.
Probabilities of solving and of not solving:
\( \displaystyle \text{Kashvi: } \frac{2}{5},\ \frac{3}{5} \qquad \text{Kalyani: } \frac{1}{4},\ \frac{3}{4} \qquad \text{Sara: } \frac{1}{6},\ \frac{5}{6} \)
Since they work independently, the probabilities multiply:
\( \displaystyle P(\text{only Kashvi}) = \frac{2}{5}\times\frac{3}{4}\times\frac{5}{6} = \frac{30}{120} \)
\( \displaystyle P(\text{only Kalyani}) = \frac{3}{5}\times\frac{1}{4}\times\frac{5}{6} = \frac{15}{120} \)
\( \displaystyle P(\text{only Sara}) = \frac{3}{5}\times\frac{3}{4}\times\frac{1}{6} = \frac{9}{120} \)
These three cases cannot happen together, so their probabilities add:
\( \displaystyle P(\text{exactly one}) = \frac{30+15+9}{120} \)
\( \displaystyle = \frac{54}{120} = \frac{9}{20} \)