Three students, A, B and C can respectively solve 50%, 25% and 20% of the problems in a book. A particular problem is selected at random from the book. The probability that at least one of them will solve the problem is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{7}{10}\)
The three students try the problem independently (the usual reading of such questions):
\( \displaystyle P(A) = \frac{1}{2},\quad P(B) = \frac{1}{4},\quad P(C) = \frac{1}{5} \)
“At least one solves it” is the complement of “none solves it”:
\( \displaystyle P(\text{none}) = \frac{1}{2}\times\frac{3}{4}\times\frac{4}{5} = \frac{3}{10} \)
\( \displaystyle P(\text{at least one}) = 1 - \frac{3}{10} = \frac{7}{10} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{20}\) — Incorrect. The probability is \(1 - \frac{3}{10} = \frac{7}{10}\), not \(\frac{3}{20}\).
Option (2) \(\displaystyle\frac{17}{20}\) — Incorrect. The probability is \(\frac{7}{10} = \frac{14}{20}\), not \(\frac{17}{20}\).
Option (4) \(\displaystyle\frac{3}{10}\) — Incorrect. This is the probability that none of them solves the problem, the complement of the event asked.