Three students P, Q and R try a puzzle independently of one another. Their probabilities of solving it are \(\displaystyle\frac{1}{2}\), \(\displaystyle\frac{1}{3}\) and \(\displaystyle\frac{1}{5}\) respectively. Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | Probability that the puzzle is solved | (I) | \(\displaystyle\frac{4}{15}\) |
| (B) | Probability that exactly two of them solve it | (II) | \(\displaystyle\frac{1}{30}\) |
| (C) | Probability that all three of them solve it | (III) | \(\displaystyle\frac{11}{15}\) |
| (D) | Probability that none of them solves it | (IV) | \(\displaystyle\frac{7}{30}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
P, Q and R fail with probabilities \(\frac{1}{2}\), \(\frac{2}{3}\) and \(\frac{4}{5}\). They work independently, so the chance of each combination is a product.
(A) Probability that the puzzle is solved
This matches (III) \(\frac{11}{15}\).
The puzzle stays unsolved only when all three fail:
\( \displaystyle P(\text{solved}) = 1-\frac{1}{2}\times\frac{2}{3}\times\frac{4}{5} = 1-\frac{4}{15} = \frac{11}{15} \)
(B) Probability that exactly two of them solve it
This matches (IV) \(\frac{7}{30}\).
\( \displaystyle P(\text{P, Q only}) = \frac{1}{2}\times\frac{1}{3}\times\frac{4}{5} = \frac{4}{30} \)
\( \displaystyle P(\text{P, R only}) = \frac{1}{2}\times\frac{2}{3}\times\frac{1}{5} = \frac{2}{30} \)
\( \displaystyle P(\text{Q, R only}) = \frac{1}{2}\times\frac{1}{3}\times\frac{1}{5} = \frac{1}{30} \)
\( \displaystyle P(\text{exactly two}) = \frac{4+2+1}{30} = \frac{7}{30} \)
(C) Probability that all three of them solve it
This matches (II) \(\frac{1}{30}\).
\( \displaystyle P(\text{all three}) = \frac{1}{2}\times\frac{1}{3}\times\frac{1}{5} = \frac{1}{30} \)
(D) Probability that none of them solves it
This matches (I) \(\frac{4}{15}\).
\( \displaystyle P(\text{none}) = \frac{1}{2}\times\frac{2}{3}\times\frac{4}{5} = \frac{8}{30} = \frac{4}{15} \)